BAMV-XXI-2 - ivic.gob.ve

Transcription

Boletín
Asociación Matemática Venezolana
Maryam Mirzakhani y Artur Ávila, primera mujer y primer
latinoamericano en ser galardonados con la Medalla Fields
Vol. XXI • No. 2 • Año 2014
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Boletı́n de la Asociación Matemática Venezolana
Volumen XXI, No. 2, Año 2014
I.S.S.N. 1315–4125
&
%
Editor
Oswaldo Araujo
Editores Asociados
Carlos Di Prisco y Henryk Gzyl
Editor Técnico: Neptalı́ Romero
Comité Editorial
Pedro Berrizbeitia, Alejandra Cabaña, Giovanni Calderón, Sabrina
Garbin, Gerardo Mendoza, Neptalı́ Romero, Rafael Sánchez
Lamoneda, Judith Vanegas, Jorge Vargas
El Boletı́n de la Asociación Matemática Venezolana se publica dos veces al
año en forma impresa y en formato electrónico. Sus objetivos, información para
los autores, direcciones postal y electrónica se encuentran en el interior de la
contraportada. Desde el Volumen VI, Año 1999, el Boletı́n aparece reseñado en
Mathematical Reviews, MathScinet y Zentralblatt für Mathematik.
Asociación Matemática Venezolana
Presidente
Pedro Berrizbeitia
Capı́tulos Regionales
CAPITAL
Pedro Berrizbeitia
USB
[email protected]
LOS ANDES
Oswaldo Araujo
ULA
[email protected]
CENTRO-OCCIDENTAL
Alexander Carrasco
UCLA
[email protected]
ORIENTE
Said Kas-Danouche
UDO
[email protected]
ZULIA-FALCON
Oswaldo Larreal
LUZ
[email protected]
La Asociación Matemática Venezolana fue legalmente fundada en 1990 como
una organización civil cuya finalidad es trabajar por el desarrollo de la matemática en Venezuela. Para más información ver su portal de internet:
http://www.ciens.ucv.ve/ciens/amv
Boletı́n
de la
Asociación
Matemática
Venezolana
Vol. XXI • No. 2 • Año 2014
Editorial
67
La Medalla Fields 2014
Oswaldo Araujo
En 1924 en la ciudad de Toronto, Canadá, se celebró el VII International
Congress of Mathematicians (ICM). El matemático canadiense, John Charles
Fields, que fue secretario de ese ICM propuso la creación de un premio que
reconociera el trabajo relevante de jóvenes matemáticos. En 1932, durante la
celebración del IX ICM en Zurich, se aceptó su propuesta. Los primeros premios
fueron otorgados en 1936 con fondos donados por el profesor Fields.
La Medalla Internacional para Descubrimientos Sobresalientes en Matemáticas, mejor conocida como Medalla Fields en honor a su promotor, es un una distinción que concede la International Mathematical Union (IMU) a matemáticos
menores de 40 años, cada cuatro años, en reconocimiento de sus logros excepcionales en matemática. Este año el ICM se celebró en Seúl y fueron distinguidos
con el premio los siguientes matemáticos: Artur Ávila, Manjul Bhagarva, Martin
Hairer y Maryam Mirzakhani.
A continuación un breve comentario sobre ellos: Artur Ávila, matemático
brasileño-francés del CNRS y del Instituto de Matemática Pura e Aplicada
(IMPA). Su área de trabajo se centra en sistemas dinámicos y teorı́a espectral.
Ha sido galardonado con la Medalla Fields 2014 por “sus profundas contribuciones a la teorı́a de los sistemas dinámicos, que han cambiado la faz de ese campo,
utilizando la poderosa idea de la “renormalización” como principio unificador”.
Es el primer latinoamericano en ser laureado con esa medalla.
Manjul Bhagarva, matemático indo-canadiense-estadounidense de la Universidad de Princeton. Su especialidad es la teorı́a de números. El jurado le otorga
la Medalla Fields 2014 por haber desarrollado “nuevas y poderosos métodos en
geometrı́a de números”.
Martin Hairer, matemático austriaco que trabaja en la Universidad de Warwick. Su área es el análisis estocástico, es laureado por sus “excepcionales contribuciones a la teorı́a de ecuaciones en derivadas parciales estocásticas”.
Maryam Mirzakhani, investigadora de la Universidad de Stanford, especialista en teorı́a de Teichmüller, geometrı́a hiperbólica, teorı́a ergódica y geometrı́a
simpléctica. Recibe la medalla por “sus avances sobresalientes en la dinámica
y geometrı́a de las superficies de Riemann y sus espacios modulares”. Es la
primera mujer en ser galardonada con tan prestigioso premio.
El ICM 2014 marca un hito en la historia de la Medalla Fields por dos
hechos extraordinarios: primera vez que la medalla es otorgada a una mujer y,
también, por primera vez la recibe un latinoamericano. No es esta la primera
ocasión en que este par de brillantes matemáticos comparten honores; en 1995
ambos ganaron Medalla de oro en la Olimpiada Internacional de Matemáticas.
¿Cuál es la relevancia social del reconocimiento a este par de talentosos
matemáticos?
En el caso de Maryam Mirzakhani es suficiente con recordar las luchas sociales y polı́ticas que la sociedad mundial ha emprendido en pro de la igualdad
Editorial
68
de la mujer. Luchas que se remontan al siglo VI, aC, con la griega Temeo, considerada la primera matemática de la Historia. Mujeres, como la francesa Sophie
Germain (1776-1831), que no pudo ingresar a L’Ecole Polytechnique y tuvo que
publicar sus trabajos matemáticos con nombre de hombre o, la alemana Emmy
Amalie Noether (1882-1935), quién en 1905 obtuvo permiso para inscribirse en
Erlanger, donde se doctoró en 1907, pero que no consiguió una plaza como
profesora e investigadora en la universidad ¡por ser mujer!
En el premio a Artur Ávila, formado matemáticamente en un paı́s en vı́as
de desarrollo, hay un reconocimiento también a una institución: el Instituto de
Matemática Pura e Aplicada (IMPA).
El IMPA fundado en 1952 es un instituto público que, desde el 2.000, funciona como una institución privada lo que le da una gran flexibilidad y autonomı́a
para desarrollar sus programas y proyectos. Este centro es el resultado de la
determinación y trabajo de un grupo de matemáticos, entre los que hay que
mencionar a Leopoldo Nachbin, Elon Lages Lima, Mauricio Peixoto, Manfredo do Carmo y Jacob Palis Junior, de establecerse en Brasil para producir y
enseñar matemática.
Desde muy joven el medallista Ávila frecuentó el IMPA; primero participando del programa de captación de niños talentosos vı́a Olimpiada matemática,
después como estudiante de maestrı́a y, posteriormente, de doctorado, el que
obtuvo con la dirección de Wellington de Melo, un ex alumno de doctorado de
Jacob Palis, lo que ejemplifica el carácter de Escuela que tiene el IMPA.
Felicitaciones a todos los ganadores de la Medalla Fields 2014, a la IMU, a las
instituciones donde estos cuatro jóvenes realizan su trabajo y a los organizadores
del ICM 2014.
Boletı́n de la Asociación Matemática Venezolana, Vol. XXI, No. 2 (2014)
69
ARTÍCULOS
On Almost Pseudo Quasi-Conformally
Symmetric Manifolds
Hülya Bağdatlı Yılmaz
Abstract. The object of this paper is to study almost pseudo
quasi-conformally symmetric manifolds. The existence of an almost
pseudo quasi-conformally symmetric manifold is shown by two nontrivial examples. Among others we study an almost pseudo quasiconformally symmetric spacetime.
2010 Mathematics Subject Classification: 53B35.
Keywords and Phrases: Almost pseudo symmetric manifold, Quasi conformally
curvature tensor, Almost pseudo quasi-conformally symmetric manifold, Einstein
field equations.
Resumen. El objeto de este trabajo es estudiar casi seudo cuasiconformes variedades simétricas. La existencia de una casi seudo
cuasi-conformes variedades simétrica se muestra mediante dos ejemplos no triviales. Entre otros estudiamos un casi seudo cuasiconforme espacio tiempo simétrico.
1
Introduction
Spaces admitting some sense of symmetry play an important role in general
relativity. Without imposing a symmetry condition, solving the Einstein’s field
equations of gravitation becomes a difficult, if not impossible, task. The condition of conformal symmetry is one of the conditions utilized in the theoretical
development of the subject.
Let (M, g) be a Riemannian manifold of dimension n and let ∇ be its LeviCivita connection. If a curvature tensor R of (M, g) is parallel with respect
to its Levi-Civita connection ∇, namely ∇R = 0, then this manifold is called
locally symmetric [3]. A generalization of the notion of local symmetry is given
by the notion of conformal symmetry. A manifold (M, g) is said to be conformally symmetric if its conformal curvature tensor C is parallel with respect to
its Levi-Civita connection of (M, g), i.e., if ∇C = 0 [5] . The class of conformally symmetric manifolds contains all locally symmetric as well as conformally
70
flat manifolds of dimension n ≥ 4. Conformally symmetric manifolds that are
neither conformally flat nor locally symmetric were introduced by Roter [14].
Thence, conformal symmetry is a proper generalization of local symmetry.
The notion of locally symmetric manifolds has been weakened by many authors in several ways to a different extent such as conformally symmetric manifolds by Chaki and Gupta [5], recurrent manifolds introduced by Walker [19],
conformally recurrent manifolds by Adati and Miyazawa [1], pseudo symmetric
manifolds introduced by Chaki [4], weakly symmetric manifolds by Tamassy and
Binh [18], projective symmetric manifolds introduced by Soos [16], projectivesymmetric and projectively recurrent affinely connected spaces by Mikeš [12],
pseudo conformally symmetric spaces by De and Biswas [7], almost pseudo
conformally symmetric manifolds introduced by De and Gazi [8], weakly conformally symmetric manifolds by De and Bandyopadhyay [6], etc.
A non-flat Riemannian manifold (M, g) (n ≥ 2) is called an almost pseudo
symmetric manifold whose curvature tensor R of type (0, 4) satisfies the condition [8]
(∇X R) (Y, E, U, V )
=
[A(X) + B(X)] R(Y, E, U, V )
+A(Y )R(X, E, U, V ) + A(E)R(Y, X, U, V ) (1.1)
+A(U )R(Y, E, X, V ) + A(V )R(Y, E, U, X),
where A and B are nowhere vanishing 1-forms, such that
A(X) = g(X, ρ) and B(X) = g(X, Q)
(1.2)
for all X and ρ and Q are the vector fields associated with the 1-forms A and
B, respectively. An n-dimensional almost pseudo symmetric manifold has been
denoted by A(P S)n . De and Gazi [8] further showed the physical significance
in general relativity and proved its existence by several examples. If A = B
in (1.1) then the manifold reduces to a pseudo symmetric manifold (P S)n [4].
The notion of pseudo symmetry in the sense of Chaki [4] is different from that
of Deszcz [10]. It is to be noted that the almost pseudo symmetric manifold is
not a special case of a weakly symmetric manifold (W S)n [18].
The notion of the quasi-conformal curvature tensor was defined by Yano and
Sawaki in 1968 [22] . According to them a quasi-conformal curvature tensor W
on a Riemannian manifold contains the conformal curvature tensor C as well
as the concircular curvature tensor Z as special cases and is defined by
W (X, Y, V, U ) = −(n − 2)bC(X, Y, V, U ) + [a + (n − 2)b] Z(X, Y, V, U ) (1.3)
for all vector fields X, Y, V, U ∈ χ(M ), where a and b are arbitrary constants,
C and Z are a conformal curvature tensor and a concircular curvature tensor
of type (0, 4) respectively.
On Almost Pseudo Quasi-Conformally Symmetric Manifolds 71
The conformal curvature tensor C of a Riemannian manifold (M, g)(n > 3)
of type (0, 4) is given by Pokharial and Misra [15]
C(X, Y, V, U )
= R(X, Y, V, U )
1
−
[S(Y, V )g(X, U ) − S(X, V )g(Y, U )
(1.4)
n−2
+ g(Y, V )S(X, U ) − g(X, V )S(Y, U )]
r
+
[g(Y, V )g(X, U ) − g(X, V )g(Y, U )] ,
(n − 1)(n − 2)
where S and r denote respectively the Ricci tensor of type (0, 2) and the scalar
curvature.
A (0, 4) type tensor Z(X, Y, V, U ) which remains invariant under concircular
transformation, for a Riemannian manifold (M, g), is given by Yano and Kon
[20, 21]
Z(X, Y, V, U ) = R(X, Y, V, U )
−
r
[g(Y, V )g(X, U ) − g(X, V )g(Y, U )]
n(n − 1)
(1.5)
where r is the scalar curvature. The importance of concircular curvature tensor
is very well known in the differential geometry of certain F-structure such as
complex, almost complex, Kahler, almost Kahler, contact and almost contact
structure etc. [2, 17]. In a recent paper Ahsan and Siddiqui [1] studied the
application of concircular curvature tensor in fluid spacetime.
The present paper is concerned with a non quasi-conformally flat Riemannian manifold (M, g) (n > 3) whose quasi-conformally curvature tensor W
satisfies the condition
(∇X W ) (Y, E, U, V ) = [A(X) + B(X)] W (Y, E, U, V )
+ A(Y )W (X, E, U, V ) + A(E)W (Y, X, U, V )
+ A(U )W (Y, E, X, V ) + A(V )W (Y, E, U, X), (1.6)
where A and B have the meaning already stated. Such a manifold will be
called an almost pseudo quasi-conformally symmetric manifold and denoted
by A(P W S)n , where W stands for ‘quasi-conformal curvature tensor’. Since
the conformal curvature tensor vanishes identically for n = 3, we assume the
condition n > 3 throughout the paper.
The present paper is organized as follows: After preliminaries in section
3, it is proved that in an A(P W S)n , nr is an eigenvalue of the Ricci tensor
S corresponding to the eigenvector ρ defined in (1.2) provided that a + (n −
2)b 6= 0. However, we obtained a sufficient condition for an A(P W S)n to be
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an A(P S)n . Section 4 deals with an Einstein A(P W S)n . In section 5, nontrivial examples of an A(P W S)n have been constructed. Finally, we study an
A(P W S)n spacetime.
2
Preliminaries
Let L be the symmetric endomorphism of the tangent space at any point of
the manifold corresponding to the Ricci tensor S of type (0, 2), such that
S(X, Y ) = g(LX, Y )
(2.1)
Let {ei } , (1 ≤ i ≤ n) be an orthonormal basis of the tangent space at any point
of the manifold.
Now, let us put X = U = ei in (1.4) and (1.5) and take summation over i
(1 ≤ i ≤ n), we then get
e
C(Y,
V ) = 0,
(2.2)
and
r
g(Y, V ),
(2.3)
n
n
n
P
P
e
where C(Y,
V) =
C(ei , Y, V, ei ) and P (Y, V ) =
Z(ei , Y, V, ei ), respecP (Y, V ) = S(Y, V ) −
i=1
i=1
tively. In virtue of (2.3), we have
n
X
P (ei , ei ) = 0.
(2.4)
i=1
Taking X = U = ei in (1.3) and using (2.2) and (2.3), we have
n
X
W (ei , Y, V, ei ) = [a + (n − 2)b] P (Y, V ).
(2.5)
i=1
Let us now write (1.6) in local coordinates, which gives the following form
∇i Wklmj = (Ai + Bi ) Wklmj +Ak Wilmj +Al Wkimj +Am Wklij +Aj Wklmi . (2.6)
From (2.5), we can write
r
Wlm = [a + (n − 2)b] Plm = [a + (n − 2)b] Slm − glm .
n
(2.7)
Transvecting (2.6) with g kj , we get
j
k
∇i Wlm = (Ai + Bi ) Wlm + Ak Wilm
+ Al Wim + Am Wli + Aj Wlmi
.
(2.8)
Transvecting (2.8) with g lm and using (2.7) and (2.4), we obtain
lm
k
k
Ak Wilm
+ Wlmi
g = 0.
(2.9)
From (2.7), it follows that
r 2 [a + (n − 2)b] Ak Sik − δik = 0.
n
(2.10)
It follows from (2.10) that
2 [a + (n − 2)b] P (X, ρ) = 0.
(2.11)
Therefore we get
P (X, ρ) = 0
unless a + (n − 2)b = 0.
(2.12)
From (2.3) and (2.12), it follows that
S(X, ρ) =
3
r
g(X, ρ).
n
(2.13)
Nature of scalar curvature r of an A(P W S)n
From (2.13) we can state the following:
Theorem 1. In an A(P W S)n , if a + (n − 2)b 6= 0 then nr is an eigenvalue of
the Ricci tensor S corresponding to the eigenvector ρ defined in (1.2).
We now assume that the scalar curvature r of an A(P W S)n vanishes, then
from (2.13) it follows that S(X, ρ) = 0. Substitution of X by Y yields
S(Y, ρ) = 0.
(3.1)
Thus from (1.3), (1.2), (1.4) and (1.5) we get
W (X, Y, ρ, U ) = aR(X, Y, ρ, U ) + b [A(Y )S(X, U ) − A(X)S(Y, U )] .
(3.2)
Hence we can state the following theorem:
Theorem 2. On the condition a + (n − 2)b 6= 0, the quasi-conformal curvature
tensor W of an A(P W S)n with zero scalar curvature is as in (3.2).
Adversely, let W be the form (3.2). Then from (1.3) it follows that
a + 2(n − 1)b
r [A(Y )g(X, U ) − A(X)g(Y, U )] = 0.
n(n − 1)
(3.3)
Since A is the nowhere vanishing 1-form, if a + 2(n − 1)b 6= 0, then we get r = 0.
Thus we have the following theorem:
74
Theorem 3. If the conditions (3.2) and a + (n − 2)b 6= 0 are satisfied in an
A(P W S)n , then it is of zero scalar curvature provided that a + 2(n − 1)b 6= 0 .
Substituting (1.4) and (1.5) in (1.3), we obtain
W (X, Y, V, U )
=
aR(X, Y, V, U )
(3.4)
+b [S(Y, V )g(X, U ) − S(X, V )g(Y, U )
+g(Y, V )S(X, U ) − g(X, V )S(Y, U )]
r
a
−
+ 2b [g(Y, V )g(X, U )
n n−1
−g(X, V )g(Y, U )] .
Let us assume that the Ricci tensor S of an A(P W S)n vanishes. Then from
(3.4), it follows that
W (X, Y, V, U ) = aR(X, Y, V, U ).
(3.5)
Thus, by virtue of (1.6), for a 6= 0 we obtain a sufficient condition for an
A(P W S)n to be an A(P S)n . This can be stated as the following:
Theorem 4. If in an A(P W S)n the Ricci tensor vanishes, then A(P W S)n
reduces to an A(P S)n on the condition a 6= 0.
4
Einstein A(P W S)n
We consider an A(P W S)n defined by (1.6), which is an Einstein manifold. Then
its Ricci tensor S satisfies
S(Y, V ) =
r
g(Y, V ).
n
(4.1)
From which it follows that
dr(Y ) = 0 and (∇X S) (Y, V ) = 0.
(4.2)
Due to (4.1) and (3.4) we get
W (X, Y, V, U ) = aR(X, Y, V, U )
ar
−
[g(Y, V )g(X, U ) − g(X, V )g(Y, U )] .
n(n − 1)
(4.3)
Taking the covariant derivative with respect to E on both sides, we obtain
(∇E W ) (X, Y, V, U ) = a (∇E R) (X, Y, V, U ).
(4.4)
Using (1.3), (1.6) and (4.3), we have
a (∇E R) (X, Y, V, U ) = −(n − 2)b [(A(E) + B(E)) C(X, Y, V, U )
+ A(X)C(E, Y, V, U ) + A(Y )C(X, E, V, U )
+A(V )C(X, Y, E, U ) + A(U )C(X, Y, V, E)]
+ [a + (n − 2)b] [(A(E) + B(E)) Z(X, Y, V, U )
+ A(X)Z(E, Y, V, U ) + A(Y )Z(X, E, V, U )
+A(V )Z(X, Y, E, U ) + A(U )Z(X, Y, V, E)] . (4.5)
Since an A(P W S)n is an Einstein space, its conformal curvature tensor forms
the following
C(X, Y, V, U ) = R(X, Y, V, U )
r
−
[g(Y, V )g(X, U ) − g(X, V )g(Y, U )] .
n(n − 1)
(4.6)
This means that the conformal curvature tensor C equals the concircular curvature tensor Z. Thus from (4.5) and (4.6) provided that a 6= 0, it follows
that
(∇E R) (X, Y, V, U )
=
[(A(E) + B(E)) C(X, Y, V, U )
+A(X)C(E, Y, V, U ) + A(Y )C(X, E, V, U ) (4.7)
+A(V )C(X, Y, E, U ) + A(U )C(X, Y, V, E)] .
Substituting (4.6) in (4.7), we obtain
(∇E R)(X, Y, V, U )
=
[A(E) + B(E)]{R(X, Y, V, U )
r
−
[g(Y, V )g(X, U ) − g(X, V )g(Y, U )]}
n(n − 1)
+A(X){R(E, Y, V, U )
r
−
[g(Y, V )g(E, U ) − g(E, V )g(Y, U )]}
n(n − 1)
+A(Y ){R(X, E, V, U )
(4.8)
r
−
[g(E, V )g(X, U ) − g(X, V )g(E, U )]}
n(n − 1)
+A(V ){R(X, Y, E, U )
r
−
[g(Y, E)g(X, U ) − g(X, E)g(Y, U )]}
n(n − 1)
+A(U ){R(X, Y, V, E)
r
−
[g(Y, V )g(X, E) − g(X, V )g(Y, E)]}.
n(n − 1)
76
We now suppose that the Einstein A(P W S)n is an A(P S)n . Then in virtue
of (4.8), we get
r
{[A(E) + B(E)] [g(Y, V )g(X, U ) − g(X, V )g(Y, U )]
n(n − 1)
+A(X) [g(Y, V )g(E, U ) − g(E, V )g(Y, U )]
(4.9)
+A(Y ) [g(E, V )g(X, U ) − g(X, V )g(E, U )]
+A(V ) [g(Y, E)g(X, U ) − g(X, E)g(Y, U )]
+A(U ) [g(Y, V )g(X, E) − g(X, V )g(Y, E)]} = 0.
Substituting X = U = ei in (4.9) and taking summation over i (1 ≤ i ≤ n), it
follows that
r{[(n + 1)A(E) + (n − 1)B(E)]g(Y, V )
+ (n − 2)A(Y )g(E, V ) + (n − 2)A(V )g(Y, E)} = 0. (4.10)
Again, substituting Y = V = ei in (4.10) and taking summation over i (1 ≤ i ≤
n), then we obtain
r [(n + 4)A(E) + nB(E)] = 0,
(4.11)
and putting E = Y = ei in (4.10), it follows that
r [(n + 1)A(V ) + B(V )] = 0.
(4.12)
Replacing V by E in the above relation, we get
r [(n + 1)A(E) + B(E)] = 0.
(4.13)
Similarly putting E = V = ei in (4.10) and taking summation over i (1 ≤ i ≤ n),
we have
r [(n + 1)A(Y ) + B(Y )] = 0.
(4.14)
Substitution of Y by E in the last relation yields
r [(n + 1)A(E) + B(E)] = 0.
(4.15)
Adding (4.11), (4.13) and (4.15), we obtain
r [3A(E) + B(E)] = 0.
(4.16)
From (4.15), it follows that r = 0 if
3A(E) + B(E) 6= 0
Hence we can state the following :
(4.17)
Theorem 5. If an Einstein A(P W S)n is an A(P S)n , then its scalar curvature
is zero provided that the vector fields ρ and Q corresponding to the associated
1-forms A and B are not linearly dependent and a 6= 0.
Adversely, let r = 0 be in an Einstein A(P W S)n . Then due to (4.8), it follows
that an Einstein A(P W S)n becomes to an A(P S)n in condition of a 6= 0. Then
we state the following theorem:
Theorem 6. If the scalar curvature of an Einstein A(P W S)n vanishes, then
such a manifold reduces to an A(P S)n provided that a 6= 0.
We now assume that the vector field Q associated with 1-form B is parallel
in an Einstein A(P W S)n . Then we get
∇X Q = 0
for all X.
(4.18)
Hence we obtain
R(X, Y )Q = ∇X ∇Y Q − ∇Y ∇X Q − ∇[X,Y ] Q = 0,
(4.19)
R(X, Y, Q, U ) = 0.
(4.20)
S(Y, Q) = 0.
(4.21)
and
Contracting (4.20) we have
Now, by (4.18) and (4.21) we get
(∇X S) (Y, Q) = ∇X S(Y, Q) − S (∇X Y, Q) − S (Y, ∇X Q) = 0.
(4.22)
From (4.8) it follows that
(∇E S) (Y, V ) = A (R(E, Y )V )
−
r
[g(Y, V )A(E) − g(E, V )A(Y )] . (4.23)
n(n − 1)
Putting V = Q in (4.23) and applying (4.19) and (4.22) and (1.2) we obtain
r
[B(Y )A(E) − B(E)A(Y )] = 0.
n(n − 1)
If B(Y )A(E) − B(E)A(Y ) 6= 0, we have r = 0 and then from (4.8) we see that
the manifold becomes an A(P S)n .Hence we have the following theorem:
Theorem 7. If the vector field Q is a parallel vector field in an Einstein
A(P W S)n , then A(P W S)n reduces to an A(P S)n provided that the vector fields
ρ and Q corresponding to the associated 1-forms A and B are not codirectional
and a 6= 0.
78
5
Examples of an A(P W S)n
In this section we address some examples of an A(P W S)n . On coordinate
space Rn (with coordinates x1 , x2 , ..., xn ), we calculate the components of the
curvature tensor, the Ricci tensor , the quasi conformal curvature tensor and
its covariant derivatives. Then we verify the relation (1.6).
Example 1. Let M 5 = x1 , x2 , x3 , x4 , x5 ∈ R5 : x1 6= −1 be an open subset of R5 .We define a Riemannian metric on the M 5 by the formula [11]
ds2 = (x1 + 1)(x3 )2 (dx1 )2 + 2dx1 dx2 + (dx3 )2 + (dx4 )2 + (dx5 )2 .
(5.1)
Then the only non-vanishing components of the Christoffel symbols and the
curvature tensors are
1 3 2 3
(x ) , Γ11 = −(x1 + 1)x3 = −Γ213 ,
(5.2)
Γ211 =
2
1
R1331 = x + 1,
and the components obtained by the symmetry properties.
In the metric considered, the covariant and contravariant components of the
metric are as follows:
g11
=
(x1 + 1)(x3 )2 , g12 = g21 = 1,
g33
=
g44 = g55 = 1,
(5.3)
and
g 11
=
0, g 22 = −(x1 + 1)(x3 )2 ,
g 12
= g 21 = g 33 = g 44 = g 55 = 1.
Due to (5.2) and (5.3), the non-vanishing components of the Ricci tensor are
S11 = x1 + 1.
(5.4)
From r = g ij Sij = g 11 S11 + g 22 S22 + g 33 S33 + g 44 S44 + g 55 S55 , using (5.3) and
(5.4), it can be easily seen that the scalar curvature of (M 5 , g) is the following:
r = 0.
(5.5)
Therefore (M 5 , g) has zero scalar curvature.
Now let us calculate the quasi-conformal curvature. In virtue of (1.4) and
(1.5), we obtain that the only non-vanishing components of the conformal curvature tensor C and the concircular curvature tensor Z of (M 5 , g) are given by
the following relations:
C1331 =
2 1
(x + 1),
3
1
C1441 = C1551 = − (x1 + 1),
3
and
Z1331 = x1 + 1,
respectively. Thus from (1.3), it follows that
W1331 = (a + b)(x1 + 1) 6= 0,
W1441 = W1551 = b(x1 + 1) 6= 0.
(5.6)
Hence (M 5 , g) is non- quasi-conformally flat. From (5.6), it can be easily shown
that the only non-zero terms of ∇l Wikjm are
∇1 W1331 = a + b 6= 0, ∇1 W1441 = ∇1 W1551 = b 6= 0.
(5.7)
All other components of ∇l Wikjm vanish identically. Thus our M 5 with the
considered metric g in (5.1) is a Riemannian manifold with a vanishing scalar
curvature that is neither quasi-conformally symmetric nor quasi-conformally
flat.
In terms of the local coordinate system, we consider the components of the
1-forms A and B as follows:
1
1
for i = 1
6(x1 +1)
2(x1 +1) , for i = 1
, Bi =
.
(5.8)
Ai =
0
otherwise
0
otherwise
In (M 5 , g) the considered 1-forms (1.6) to reduce to the following equations:
1. ∇1 W1331 = (3A1 + B1 ) W1331 + A3 W1131 + A3 W1311
2. ∇3 W1131 = (2A3 + B3 ) W1131 + A1 W3131 + A1 W1331 + A1 W1133
3. ∇3 W1311 = (2A3 + B3 ) W1311 + A1 W3311 + A1 W1331 + A1 W1313
4. ∇1 W1441 = (3A1 + B1 ) W1441 + A4 W1141 + A4 W1411
5. ∇4 W1141 = (2A4 + B4 ) W1141 + A1 W4141 + A1 W1441 + A1 W1144
6. ∇4 W1411 = (2A4 + B4 ) W1411 + A1 W4411 + A1 W1441 + A1 W1414
7. ∇1 W1551 = (3A1 + B1 ) W1551 + A5 W1151 + A5 W1511
8. ∇5 W1151 = (2A5 + B5 ) W1151 + A1 W5151 + A1 W1551 + A1 W1155
9. ∇5 W1511 = (2A5 + B5 ) W1511 + A1 W5511 + A1 W1551 + A1 W1515 ,
since, for the case other than (1) - (9), the components of each term of (1.6)
either vanishes identically or the relation (1.6) holds trivially. It can be easily seen that the equations (1)-(9) hold. Thus (M 5 , g) is an A(P W S)5 with
vanishing scalar curvature.
Hence we can state the following:
80
Theorem 8. Let M 5 = x1 , x2 , x3 , x4 , x5 ∈ R5 : x1 6= −1 be an open subset of R5 equipped with the metric
ds2 = (x1 + 1)(x3 )2 (dx1 )2 + 2dx1 dx2 + (dx3 )2 + (dx4 )2 + (dx5 )2 .
Then (M 5 , g) is an A(P W S)5 with vanishing scalar curvature that is neither
quasi-conformally symmetric nor quasi-conformally flat.
Example 2. Let each Latin index run over 1, 2, ..., n and each Greek index
run over 2, 3, .., n − 1. We define a Riemannian metric on the Rn , (n ≥ 4) by
the formula,
2
ds2 = ϕ dx1 + kαβ dxα dxβ + 2dx1 dxn ,
(5.9)
where [kαβ ] is a symmetric and non-singular matrix consisting of constants
and ϕ is independent of xn . In the metric considered, the only non-vanishing
components of the Christoffel sembols and the curvature tensor and Ricci tensor
are in the following [14, 9]:
Γβ11
=
R1αβ1
=
ϕ.1
1
Γn11 =
,
− k αβ ϕ.α ,
2
2
1
1
ϕ.αβ , S11 = k αβ ϕ.αβ ,
2
2
Γn1α =
ϕ.α
,
2
(5.10)
respectively, where
(.)
denotes the partial differentation with respect to the
coordinates and k αβ is the inverse matrix of [kαβ ]. For the metric (5.9), we
consider kαβ as δαβ and
ϕ = (Mαβ + δαβ )xα xβ (x1 )2/3 ,
(5.11)
where Mαβ are constants and satisfy the following relations:
Mαβ
= 0,
6= 0,
for α 6= β
for α = β
and
n−1
X
Mαα = 0,
(5.12)
α=2
[9]. Thus from (5.11) and (5.12), it follows that
ϕ.αβ = 2(Mαβ + δαβ )xα xβ (x1 )2/3 , δαβ δ αβ = n − 2 and
δ αβ Mαβ =
n−1
X
Mαα = 0. (5.13)
α=2
Therefore due to (5.10), (5.11) and (5.12), the only non-zero components of the
curvature tensor and the Ricci tensor are
R1αα1 = (1 + Mαα )(x1 )2/3 , S11 = (n − 2)(x1 )2/3 .
(5.14)
In the metric considered, the covariant and contravariant components of the
metric tensor are in the following:
g11
=
ϕ,
g 11
=
0, g αα = 1, g 1n = g n1 = 1, g nn = −ϕ.
gαα = 1,
g1n = gn1 = 1,
(5.15)
In virtue of (5.14) and (5.15), it can be easily seen that the scalar curvature
is zero. From (1.4) and (1.5), it can be shown that the only non-vanishing
components of the conformal curvature tensor C and the concircular curvature
tensor Z are
C1αα1 = Mαα (x1 )2/3
and
Z1αα1 = (1 + Mαα )(x1 )2/3 ,
(5.16)
which never vanish.
We shall now show that (Rn , g) is an A(P W S)n . It follows from (1.3) that
non-vanishing components of the quasi-conformal curvature tensor W are as
follows:
W1αα1 = [(1 + Mαα )a + (n − 2)b] (x1 )2/3 6= 0.
(5.17)
From (5.17), it can be easily shown that the only non-zero terms of ∇l Wikjm
are
2 [(1 + Mαα )a + (n − 2)b]
6= 0.
(5.18)
∇1 W1αα1 =
3(x1 )1/3
Hence Rn equipped with the metric considered is neither quasi-conformally
symmetric nor quasi-conformally flat.
In terms of the local coordinate system, we consider the components of the
1-forms A and B as follows:
1
1
i=1
for
i=1
9x1 for
3x1
Ai =
and
Bi =
.
0
for
i 6= 1
0 for
i 6= 1
In (Rn , g) with the considered 1-forms, the relation (1.6) reduces to the following
equations :
1. ∇1 W1αα1 = (3A1 + B1 ) W1αα1 + Aα W11α1 + Aα W1α11
2. ∇α W11α1 = (2Aα + Bα ) W11α1 + A1 Wα1α1 + A1 W1αα1 + A1 W11αα
3. ∇α W1α11 = (2Aα + Bα ) W1α11 + A1 Wαα11 + A1 W1αα1 + A1 W1α1α ,
since, for the case other than (1), (2) and (3), the components of each term of
(1.6) either vanishes identically or the relation (1.6) holds trivially. It can be
easily seen that the equations (1)-(3) hold. Therefore Rn under consideration
is an A(P W S)n with vanishing scalar curvature, and hence we can state the
following:
82
Theorem 9. Let Rn (n ≥ 4) be a Riemannian manifold equipped with the
metric
2
(2 ≤ α, β ≤ n − 1)
ds2 = ϕ dx1 + kαβ dxα dxβ + 2dx1 dxn ,
ϕ =
(Mαβ + δαβ )xα xβ (x1 )2/3 ,
where Mαβ are constants defined in (5.12). Then (Rn , g) is an A(P W S)n
with vanishing scalar curvature that is neither quasi-conformally symmetric nor
quasi-conformally flat.
6
A(P W S)n Spacetime
In general relativity the matter content of spacetime is described by the energy
momentum tensor T which is to be determined from physical considerations
dealing with the distribution of matter and energy. Since the matter content
of the universe is assumed to behave like a perfect fluid in the standart cosmological models, the physical motivation for studying Lorentzian manifolds is
the assumption that a gravitational field may be effectively modeled by some
Lorentzian metric defined on a suitable four dimensional manifold M .
In this section we want to deal with the study of A(P W S)n , in general
relativity by the coordinat free method of differential geometry. In this method
of study the spacetime of general relativity is regarded as a connected fourdimensional semi-Riemannian manifold (M, g) with Lorentz metric g with signature (−, +, +, +) .
Here we consider a perfect fluid A(P W S)4 spacetime of non-zero scalar
curvature and having the basic vector field ρ as the timelike vector field of the
fluid, that is, g(ρ, ρ) = −1.
For the perfect fluid spacetime, we have the Einstein equation without cosmological constant as
1
S(X, Y ) − rg(X, Y ) = kT (X, Y ),
2
(6.1)
where k is the gravitational constant, T is the energy momentum tensor of type
(0, 2) given by [13]
T (X, Y ) = (σ + p)A(X)A(Y ) + pg(X, Y ),
(6.2)
with σ and p the energy density and the isotropic pressure of the fluid respectively, A is a non-zero 1-form defined by g(X, ρ) = A(X) for all X , ρ being the
velocity vector field of the fluid.
The Einstein equations are fundamental in the construction of cosmological
models which imply that the matter determines the geometry of the spacetime
and conversely the motion of matter is determined by the metric tensor of the
space which is non- flat.
It is noted that the basic geometric features of A(P W S)n are also being
maintained in the Lorentzian manifold which is necessarily a semi-Riemannian
manifold. Thus Theorem 1 is also true for an A(P W S)4 spacetime. From (6.1)
and (6.2), it follows that
1
S(X, Y ) − rg(X, Y ) = k [(σ + p)A(X)A(Y ) + pg(X, Y )] .
2
(6.3)
Substituting Y = ρ in (6.3) and using Theorem 1 and g(ρ, ρ) = A(ρ) = −1
yields
r
σk −
A(X) = 0.
(6.4)
4
r
Since k 6= 0 and A 6= 0 , from (6.4) it follows that σ = 4k
. We know that
pure matter exists in case of σ > 0. Hence for non-zero scalar curvature ,the
spacetime under consideration can contain pure matter.
Now we deal with an A(P W S)4 , perfect fluid spacetime of non- zero scalar
curvature with velocity vector fluid ρ obeying Einstein’s equation with cosmological constant. Then Einstein’s equation is given by
1
S − rg + λg = k [(σ + p) A ⊗ A + pg] ,
2
(6.5)
where λ is the cosmological constant.
The equation (6.5) can be expressed in the following form
1
S(X, Y ) − rg(X, Y ) + λg(X, Y ) = k [(σ + p) A(X)A(Y ) + pg(X, Y )] . (6.6)
2
Putting Y = ρ in (6.6) and using Theorem 1 and g(ρ, ρ) = A(ρ) = −1, we
obtain
h
i
r
λ − + kσ A(X) = 0.
4
From A(X) 6= 0 it follows that
σ=
r − 4λ
.
4k
(6.7)
Again taking a frame field and contracting (6.6) over X and Y we get by using
(6.7)
4λ − r
p=
.
(6.8)
4k
Thus we have the following:
84
Theorem 10. If a perfect fluid A(P W S)4 spacetime of non-zero constant scalar
curvature obeys Einstein’s equation with cosmological constant, then the pressure
and density of the fluid can be constant.
By virtue of (6.7) and (6.8) we obtain
σ + ρ = 0.
This leads the following:
Theorem 11. If a perfect fluid A(P W S)4 spacetime of non-zero scalar curvature obeys Einstein’s equation with cosmological constant, then the matter
content can not be a perfect fluid with σ + ρ 6= 0.
References
[1] Adati T. and Miyazawa T. On a Riemannian space with recurrent conformal curvature, Tensor (N. S.) 18 (1967), 348-354.
[2] Blair D. E., Kim J. S. and Tripathi M. M. On the concircular curvature
tensorof a contact metric manifold, J. Korean Math. Soc. 42(5) (2005),
883-892.
[3] Cartan, E. Sur une classe remarquable d’espacesde Riemann, Bull. Soc.
Math. France 54 (1926), 214-264.
[4] Chaki M. C. On pseudo symmetric manifolds, Analele Stiint, Univ. Al-I.
Cuza 33 (1987), 53-58.
[5] Chaki M. C. and Gupta B. On conformally symmetric spaces, Indian J.
Math. 5 (1963), 113-122.
[6] De U. C. and Bandyopadhyay S. On weakly conformally symmetric spaces,
Publi. Math. Debrecen, 57 (2000), 71-78.
[7] De U. C. and Biswas H. A. On pseudo-conformally symmetric manifolds.
Bull. Calcutta Math. Soc. 85, 5 (1993), 479-486.
[8] De U. C. and Gazi A. K. On almost pseudo symmetric manifolds, Ann.
Univ. Sci. Budapest. Eötvös, Sect. Math. 51 (2008), 53-68.
[9] De U. C. and Gazi A. K. On conformally flat almost pseudo Ricci symmetric
manifolds, Kyungpook Math. 49 (2009), 507-520.
[10] Deszcz R. On pseudo symmetric spaces, Bull. Soc. Math. Belg. Serie A 44
(1992), 1-34.
[11] Hui S. K., Matsuyama Y. and Shaikh A. A. On decomposable weakly
conformally symmetric manifolds, Acta Math. Hungar. 128(1-2) (2010),
82-85.
[12] Mikeš, J., Projective-symmetric and projective-recurrent affinely connected
spaces. Tr. Geom. Semin. 13 (1981), 61–62 (in Russian).
[13] O’Neill B. Semi Riemannian geometry with applications to relativity. Academic Press, New York-London (1983).
[14] Roter W. On conformally symmetric Ricci recurrent spaces, Colloq. Math.
31 (1974), 87-86.
[15] Pokharial G.P. and Misra R.S. Curvature tensors and their relativistic significance, Yokohama Math. Jour. 18(2) (1970), 105-108.
[16] Soos G. Über die geodätischen abbildungen von Riemannschen Räumen
auf projektiv symmetrische Riemannsche Räume, Acta Math Acad. Sci.
Hungar. Tom 9 (1958), 359-361.
[17] Tashiro Y. Complete Riemannian manifolds and some vector fields, Trans.
Amer. Math. Soc. (1965), 117-251.
[18] Tamássy L. and Binh T. Q. On weakly symmetric and weakly projective
symmetric Riemannian manifolds, Colloq. Math. Soc. J. Bolyai. 50 (1989),
663-670.
[19] Wolker A. G. On Ruse’s space of recurrent curvature. Proc. London Math.
Soc. 52 (1951), 36-64.
[20] Yano K. Concircular geometry I, Imp. Acad. Sci. of Japan 16 (1940), 165.
[21] Yano K. and Kon M. Structures of manifolds, World Scientific Publishing,
Singapore. (1984).
[22] Yano K. and Sawaki S. Riemannian manifolds admitting a conformal transformation group, J. Diff. Geom. 2 (1968), 161-184.
Marmara University, Faculty of Sciences and Letters
Department of Mathematics, Istanbul- Turkey
[email protected]
87
Estabilidad y regularidad de operadores
Rosales Edixo y Klarys Fereira
Resumen. Este trabajo estudia propiedades espectrales y de regularidad de un operador. Se prueban los siguientes resultados:
• Sean X un espacio de Banach reflexivo y T ∈ B(X). Son
equivalentes: (1) r(T ) < 1 y (2) Si x ∈ X, y ∈ X 0 , entonces
{hT n x, yi}+∞
n=1 ∈ l1 (Z).
d
• Sean XTes un espacio de Banach reflexivo y T ∈ B(X). Si T n −→ 0
y σc (T ) π = ∅, entonces r(T ) < 1.
• Sea X un espacio de Banach uniformemente convexo y T ∈ B(X)
n
d
acotado por potencias. Si σ(T ) ⊂ π, Tn −→ 0 y T es un operador
lleno, entonces σp (T ) es no vacı́o; o para todo M, N ∈ latT con
N
M $ N , vale que dim M
6= 1.
Abstract. This paper studies regularly and spectral properties of
an operator. We will prove the following results:
• Let X be a reflexive Banach space and T ∈ B(X). The following
assertions are equivalents: (1) r(T ) < 1 and (2) If x ∈ X, y ∈ X 0 ,
then {hT n x, yi}+∞
n=1 ∈ l1 (Z).
d
n
• Let X
T be a reflexive Banach space and T ∈ B(X). If T −→ 0 and
σc (T ) π = ∅, then r(T ) < 1.
• Let X be a uniformly convex Banach space and T ∈ B(X) bounded
n
d
by powers. If σ(T ) ⊂ π, Tn −→ 0 y T is an full operator, then either
N
6= 1.
σp (T ) 6= ∅, or for all M, N ∈ latT with M $ N it holds dim M
1.
Preliminares
Para nosotros X será un espacio de Banach complejo y X ∗ su espacio
dual. También B(X) denotará el espacio de los operadores lineales acotados.
Si A ∈ B(X), entonces A0 será su traspuesto. Un subespacio cerrado M de X
se llama invariante para T ∈ B(X) si T M ⊂ M . Por latT entenderemos la
familia de todos los subespacios invariantes de T . Un operador T ∈ B(X) se
llamará lleno o regular, si T M = M para todo M ∈ latT , donde la barra denota la clausura topológica en la norma. Algunas veces X = H será un espacio
88
de Hilbert separable. En este caso diremos que un operador T ∈ B(H) es casi
lleno, si ∀M ∈ latT, M T M es de dimensión infinita. En general,T ∈ B(X)
será casi lleno, si para todo M ∈ latT , TMM es de dimensión infinita. Recordemos que un operador T ∈ B(X), se dirá compacto, cuando para cualquier
sucesión acotada {xn } ⊂ X, existe una subsucesión {T xnk } que es convergente.
Estos se denotan mediante K(X). Si X es un espacio de Banach, diremos que
x ∈ π, cuando kxk = 1. Es decir π es la circunferencia unitaria en el espacio de
Banach X. Dado un espacio de Banach X, y una familia de vectores {xi }i∈I ,
por [xi : i ∈ I], entenderemos el subespacio cerrado generado por dicha familia.
También si X es un espacio de Banach, diremos que {xi }+∞
i=1 es una base de
Schuader,
si
[x
:
i
∈
N
]
=
X
y
cada
x
∈
X,
se
escribe
de
manera
única como:
i
P+∞
x = i=1 αi xi (αi ∈ C). Una familia {xi }+∞
es
un
sucesión
básica,
cuando
i=1
para [xi : i ∈ N ] = M , dicha familia es una baseP
de Schauder. Recordemos que
+∞
∗
si {xi }+∞
i=1 es una base de Schuader para X, xj (
i=1 αi xi ) = αj , se llaman las
funcionales duales asociadas.
Ln
(n)
Si X es un espacio de Banach, y p > 1, entenderemos por Xp = k=1 X,
la suma directa n-veces del espacio de Banach X, con la norma: k(x1 , ..., xn )kp =
Pn
1
(n)
(n)
( k=1 kxk kp ) p . Por Tp ∈ B(X (n) ) entenderemos el operador Tp (x1 , ..., xn ) =
(T x1 , ..., T xn ). Cuando X = H es un espacio de Hilbert separable, escribiremos
(n)
simplemente X2 = H (n) . Es claro que H (n) tiene P
una estructura natural de
n
espacio de Hilbert, donde h(x1 , ..., xn ), (y1 , ..., yn )i = k=1 hxk , yk i. Recordemos
que T1 ∈ B(X1 ), se dice similar a T2 ∈ B(X2 ), cuando existe A ∈ B(X1 , X2 ),
un operador invertible, tal que A ◦ T1 ◦ A−1 = T2 .
Si T ∈ B(X), por el conmutante de T , que denotaremos por {T }0 , se entenderá la familia de todos los operadores que conmutan con T . Por AlglatT ,
representaremos la familia de todos los operadores S ∈ B(X), tales que latT ⊂
latS. Un operador T se dirá acotado por potencias, si la sucesión numérica
{kT m k}+∞
m=1 es acotada.
Una aplicación importante para nosotros es la que define J : X → X ∗∗
dada por J(x) = Jx , donde Jx (f ) = f (x), para cada f ∈ X ∗ . Es conocido que
kJ(x)k = kxk para todo x ∈ X. Si J es unitario, entonces se dirá que el espacio
X es de Banach reflexivo. Un espacio de Banach X se dirá uniformemente
convexo, si dado 0 < ≤ 2, existe δ > 0 tal que para todo x, y ∈ X, con
|xk = kyk = 1 y ≤ kx − yk se tiene k x+y
2 k ≤ 1 − δ. Todo espacio de Banach
uniformemente convexo X es reflexivo.
Recordemos que si T ∈ B(X), su espectro σ(T ) es el subconjunto de los
números complejos dado por σ(T ) = {λ ∈ C : T − λI no es invertible}. Por el
espectro puntual σp , entenderemos los λ ∈ σ(T ), tales que ker(T − λI) 6= {0}.
El espectro continuo σc (T ), lo forman los λ ∈ σ(T ), tales que (T − λI)X = X,
donde la barra denota la clausura en los números complejos C. Por la resolvente
del operador T , entenderemos ρ(T ) = C − σ(T ) y ρ∞ (T ) será la componente
no acotada de la resolvente. Un operador T ∈ B(H) se dice de radio numérico
89
alcanzable, si existe un vector unitario x ∈ H, tal que kT k = |hT x, xi|. Dado
un T ∈ B(H) sabemos que existe un T ∗ ∈ B(H) (adjunto de T ), tal que
hT x, yi = hx, T ∗ yi, ∀ x, y ∈ H. Si T ∈ B(H) es tal que hT x, xi ≥ 0, ∀ x ∈ H,
diremos que el operador es positivo. Es conocido que dado un operador positivo
T ∈ B(H), él es autoadjunto, es decir T = T ∗ y existe su raı́z cuadrada, es decir
√ 2
√
un único operador autoadjunto T ∈ B(H) tal que T = T .
Si T ∈ B(X), por T 0 ∈ B(X ∗ ), entenderemos el operador definido por
0
T (f ) = f ◦ T, ∀ f ∈ X ∗ . Dado T ∈ B(H), se dirá hiponormal si kT ∗ xk ≤
kT xk, ∀ x ∈ H. Para un T ∈ B(H), consideraremos los conjuntos numéricos:
W
T (T ) = {hT x, xi : x ∈ H, kxk = 1} (Rango numérico de T ) y We (T ) =
K∈K(X) W (T + K) (Rango numérico esencial de T ).
Si T ∈ B(X) es tal que f (T n x) → 0, ∀ f ∈ X ∗ , ∀ x ∈ X, diremos que
n
T converge débilmente al operador nulo, o que T es débilmente estable. Esto
d
lo escribiremos mediante T n −→ 0. De igual manera si kT n xk → 0, ∀ x ∈ X,
diremos que T n converge fuertemente al operador nulo. Esto lo escribiremos
f
mediante T n −→ 0. Si T ∈ B(H), por AT entenderemos el álgebra débil generada por T , donde un operador A ∈ AT , si existe una red de polinomios
{Pd (X)}, tales que h{Pd (T )x, yi → hAx, yi, ∀ x, y ∈ H. Recordemos que T es
u
uniformemente estable, si kT n k → 0, lo que escribiremos T n −→ 0.
Valen los siguientes resultados:
Teorema 1. Sea H es un espacio de Hilbert separable y T ∈ H. Si T no es
lleno, entonces existe un vector no nulo x ∈ H con hT n x, xi = 0, ∀ n ≥ 1.
Demostración. Vere referencia [2].
Teorema 2. (Sarason) Sea T ∈ B(X). Si ρ(T ) es la resolvente de T y ρ∞ es
la componente conexa no acotada de ρ(T ), entonces:
(1) Si λ y λ0 pertenecen a la misma componente conexa de ρ(T ), entonces
lat(T − λ)−1 = lat(T − λ0 )−1 .
(2) Si λ ∈ ρ∞ (T ) entonces lat(T − λ)−1 = latT .
Demostración. Ver referencia [2].
Lema 1. Si A ∈ B(X) es acotado por potencias y f ∈ X ∗ , x0 ∈ X tales que
lı́m f (Am x0 ) = α 6= 0, entonces A0 tiene un punto fijo.
m→+∞
Corolario 1. Si A ∈ B(X) es acotado por potencias,X es un espacio de Banach
reflexivo, entonces Ax = x para algún x ∈ X, x 6= 0, si y sólo si, A0 θ = θ para
algún θ ∈ X ∗ .
90
Corolario 2. Si A ∈ B(X) es acotado por potencia y A0 no tiene valores
propios de valor absoluto 1, entonces A − λI es lleno, para cada λ ∈ C, |λ| = 1.
2.
El siguiente resultado fue demostrado para espacios de Hilbert por los matemáticos Vieira, Malenbranche y Kubrusly en [8]. Bravo lo hace para espacios
de Banach separables en [3]. Nosotros presentamos una demostración para espacios de Banach reflexivos en general, siguiendo las ideas del último autor
citado.
Teorema 3. Sea X un espacio de Banach reflexivo y T ∈ B(X). Son equivalentes: (1) r(T ) < 1 y (2) Si x ∈ X, y ∈ X 0 , entonces {hT n x, yi}+∞
n=1 ∈ l1 (Z).
p
n
n
kT k < 1, entonces existen α ∈
Demostración. (1) ⇒ (2). Si r(T ) = lı́m
n→+∞
[0, 1) y n0 ∈ N, tales que ⇒ kT n k < αn , ∀ n > n0 . Tenemos que si x ∈ X, y ∈
+∞
+∞
X
X
n
∗
X , entonces
|hT x, yi| ≤ kxkkyk
αn < +∞.
n=n0
n=n0
n
(2) ⇒ (1). Si x ∈ X, y ∈ X ∗ y {hT n x, yi}+∞
n=1 ∈ l1 (Z), entonces hT x, yi → 0.
d
Es
decir T n√−→ 0. Por lo tanto existe un α > 0, tal que kT n k ≤ α ∀ n ≥ 1 ⇒
p
n
kT n k ≤ n α ⇒ r(T ) ≤ 1.
Supongamos ahora que λ ∈ C, |λ| = 1. Vamos a demostrar que λ ∈ ρ(T 0 ).
Si (T 0 − λI)y = 0 con y 6= 0, entonces T 0 y = λy. Sea x ∈ X tal que hx, yi =
6 0,
n
luego hT n x, yi = hx, (T 0 )n yi = λ hx, yi → 0. Se deduce que |λ| < 1, lo cual es
una contradicción. Es decir T 0 − λI es inyectiva.
Veamos que T 0 − λI es sobreyectiva. Si z ∈ X ∗ , consideremos la sucesión
n−1
X 1
yn =
(T 0 )j−1 z (n ≥ 2). Como
j
λ
j=1
X 1
X
m−1
m,n→+∞
m−1
0 j−1 hx, yn i−hx, ym i = hx, (T 0 )j−1 zi −
−−−−−→ 0,
hx,
(T
)
zi
≤
λj
j=n
j=n
se deduce que {yn }+∞
n=1 es débilmente de Cauchy, y como X es un espacio de
d
Banach reflexivo, entonces yn −→ 0, para algún y ∈ X ∗ .
Finalmente observemos que:
hx, ((T 0 − λI)yi =
=
=
hx, T 0 yi − λhx, yi
lı́m
n→+∞
lı́m
n→+∞
n−1
X
j=1
n−1
X 1
1
0 j
hx,
(T
)
zi
−
hx, (T 0 )j−1 zi
j−1
λj
λ
j=1
1
hx, (T 0 )n−1 zi − hx, zi = −hx, zi;
λn−1
91
por lo que (T 0 − λI)(−y) = z
Observación 1. De acuerdo al resultado anterior, si T ∈ B(H) y r(T ) <
+∞
X
1, entonces γ(x, y) =
hT n x, yi, ∀ x, y ∈ H, define una forma sesquilineal
n=1
d
continua. En efecto como r(T ) < 1, tenemos que T n −→ 0. Por lo tanto dado
0 < α < 1, existe un n0 tal que kT n k ≤ αn , ∀ n ≥ n0 . Tenemos luego que
|γ(x, y)|
≤
≤
kxkkyk
kxkkyk
n0
X
kT n k +
n=1
n0
X
+∞
X
αn
n=n0 +1
kT n k + αn0 +1 (
n=1
1 ) .
1−α
Es claro que si T es un operador positivo la forma sesquilineal γ(x, y) es positiva.
Vale el siguiente resultado:
Teorema 4. Sea T ∈ B(H) donde H es un espacio de Hilbert separable. Si
r(T ) < 1 y γ(x, y) es una forma sesquilineal positiva definida, entonces T es un
operador lleno.
Demostración. Si T no es lleno, existe un vector no nulo x ∈ H, tal que
hT n x, xi = 0, ∀ n ≥ 1. Por lo tanto γ(x, x) = 0 ⇒ x = 0, lo cual es una
contradicción.
El siguiente resultado generaliza uno dado por Kubrusly para espacios de
Hilbert en [6].
Teorema 5. Sea T ∈ B(X), donde X es un espacio de Banach reflexivo. Si
T
d
T n −→ 0 y σc (T ) π = ∅, entonces r(T ) < 1.
d
Demostración. Si T n −→ 0, entonces T es acotado por potencias y r(T ) ≤ 1. Si
|λ| = 1, entonces T − λI es inyectivo como ya lo hemos observado. Como X es
d
un espacio reflexivo, tenemos que (T 0 )n −→ 0. Veamos que T 0 no tiene valores
propios en la circunferencia unitaria. En efecto si |β| = 1 con β ∈ σp (T 0 ),
existe un funcional no nulo θ y un vector x ∈ X con θ(x) 6= 0, tales que
(T 0 )n (θ) = θ ◦ T n = β n θ ⇒ Jx ((T 0 )n (θ)) = θ ◦ T n (x) = β n θ(x) → 0 ⇒ β = 0,
lo cual es una contradicción. Usando el corolario 2, concluimos T − λI es lleno.
Se tiene que (T − λI)X = X. Si λ ∈ σ, entonces λ ∈ σc (T ), lo cual es un
contradicción, luego (T − λI)X = X ⇒ λ ∈ ρ(T ) ⇒ r(T ) < 1.
Teorema 6. Sea X un espacio de Banach uniformemente convexo, T ∈ B(X)
n
d
un operador acotado por potencias. Si σ(T ) ⊂ π, Tn −→ 0 y T es un operador
lleno, entonces σp (T ) es no vacı́o, o ∀ M, N ∈ latT con M $ N , vale que
N
dim M
6= 1.
92
N
Demostración. Supongamos que existe M, N ∈ latT tales que dim M
= 1, podemos encontrar unLvector unitario x ∈ N y un funcional unitario f ∈ X ∗ ,
tales que N = hxi M, f (x) = 1, f (M ) = 0. Como T x = βx + y, y ∈ M ,
entonces (T − βI)N ⊂ M (*). Si β = 0, entonces T N ⊂ M ⇒ T N = N ⊂ M ,
lo cual es
un contradicción. Se deduce quen β 6= 0 y por lo tanto tenemos:
n
βn
Tn
x
=
x
+ yn , yn ∈ M y ası́ f Tn x) = βn → 0 ⇒ |β| ≤ 1.
n
n
Si |β| < 1, entonces β ∈ ρ0 y por el teorema de Sarason, lat(T − βI)−1 =
latT −1 = latT ⇒ (T − βI)−1 N ⊂ N ⇒ N ⊂ (T − βI)N ⊂ M lo cual es una
contradicción. Se deduce por lo tanto que |β| = 1 y como T − βI no es un
operador lleno por (*), entonces σp (T 0 ) es no vacı́o ( ver corolario 2). Además
por ser X reflexivo, entonces por corolario 1 tenemos que σp (T ) es no vacı́o. Se
sigue el resultado.
Corolario 3. Sea X un espacio de Banach uniformemente convexo, T ∈ B(X)
invertible. Si A ∈ AlglatT es un operador acotado por potencias, tal que: σ(A) ⊂
n
d
π, An −→ 0 y A es un operador lleno; entonces σp (A) es no vacı́o, o T es un
operador lleno.
Demostración. Si σp (A) = ∅ y T no es un operador lleno, entonces existe M ∈
latT tal que dim TMM = 1. Como M, T M ∈ latA, se deduce por el resultado
anterior que dim TMM 6= 1, lo cual es una contradicción.
Teorema 7. Si H es un espacio de Hilbert y T ∈ B(H) tal que: (1) T es
f
un operador hiponormal, (2) T es de radio numérico alcanzable (3) T n −→ 0,
u
entonces T n −→ 0.
Demostración. Debemos probar que kT n k −→ 0. Como T es de radio numérico alcanzable, existe un vector unitario x, tal que kT k = |hT x, xi| =⇒ kT k ≤
kT xkkxk = kT xk =⇒ kT k = kT xk(∗), deducimos por lo tanto que x ∈ M =
{z ∈ H : kT zk = kT kkzk}. Veamos que M ∈ latT . Como kT zk = kT zkkzk,
si y sólo si, T ∗ T z = kT k2 z (Ver referencia [6, cap. 7, problema 7.5]); si suponemos que T es hiponormal y z ∈ M , entonces kT (T z)k ≤ kT kkT zk =
kT k2 kzk = kkT k2 zk = kT ∗ T zk ≤ kT (T z)k (ya que por ser T hiponormal
kT ∗ zk ≤ kT (z)k ∀ z ∈ H). Es decir kT (T z)k = kT kkT zk y por lo tanto T z ∈ M .
Veamos que kT n xk = kT kn , ∀ n ≥ 1. Sabemos por (*) que el resultado
es cierto para n = 1. Supongamos que vale para n > 1, luego kT n xk = kT kn
y como T n x ∈ M , tenemos que kT (T n x)k = kT kkT n xk = kT kkT kn lo caul
prueba lo afirmado. Al ser T hiponormal tenemos que kT kn = kT n k. Finalmente
u
kT n xk = kT kn = kT n k ⇒ T n −→ 0.
El siguiente teorema estudia propiedades de casi llenitud de operadores autoadjuntos:
93
Teorema 8. Sea H un espacio de Hilbert separable y T ∈ B(H) un operador
autoadjunto: (1) Si kerT es de dimensión finita, entonces el operador T 2 es casi
lleno. Como consecuencia todo operador T positivo tal que
√ kerT es de dimensión
finita es casi lleno.(2) Si T es positivo lleno,entonces T es lleno.
Demostración. (1) Si T es autoadjunto con kerT de dimensión finita y T 2 no
es casi lleno, entonces existe un M ∈ latT 2 , tal que M T 2 M es de dimen2
sión infinita. Sea {en }+∞
n=1 ⊂ M T M una familia ortonormal. Se deduce que
hen , T 2 en i = 0, ∀ 1 ≤ n ⇒ kT en k = 0 ⇒ en ∈ kerT , lo cual es una contradicción. Notemos que en este caso dim M T 2 M ≤ dim kerT .
√
√ 2
Si T es positivo, tenemos que kerT = ker T es finito y por lo tanto T =
T es casi lleno, lo cual prueba
lo pedido.
√
(2) Supongamos que T no es lleno, luego existe un vector no nulo x ∈ H, tal
√ 2n
√ n
que h T x, xi = 0, ∀ 1 ≤ n. Se tiene que h T x, xi = hT n x, xi = 0, ∀ n ≥ 1,
lo cual es una contradicción.
Teorema 9. Si H un espacio de Hilbert separable y T ∈ B(H), tal que 0 ∈
/
We (T ),entonces T (n) es casi lleno, ∀n ≥ 1.
Demostración. Si T no es casi lleno,entonces existe un M ∈ latT , tal que
M T M es de dimensión infinita. Sea {en }+∞
n=1 ⊂ M T M una familia ortonormal,luego hen , T en i = 0, ∀ n ≥ 1 ⇒ 0 ∈ We (T ),lo cual es una contradicción.
Como We (T (n) ) ⊂ We (T ), el resultado sale por inducción.
Lema 2. Sean X un espacio de Banach y T ∈ B(X). T no es casi lleno,si y sólo
+∞
∗
k
si, existen {fn }+∞
n=1 ⊂ X , {xn }n=1 ⊂ X, tales que fn (xm ) = δnm , fn (T xm ) =
0, ∀ n, m, k ≥ 1
Demostración. Si T no es casi lleno, existe un M ∈ latT , tal que TMM es de
M
Banach de dimensión infinita. Sea {xn + T M }+∞
n=1 una sucesión básica en T M .
Si consideramos N = [xn +T M : n ≥ 1] y las funcionales gn = (xn +T M )∗ , por
el teorema de Hahn-Banach, existe hn ∈ N ∗ que extiende a gn . Si fn = gn ◦ PN ,
donde pN : N −→ TNM es el operador proyección; entonces las extensiones de fn
al espacio de Banach X y los xn cumplen el directo.
Para ver el recı́proco, considere M = [T k xm : k ≥ 0, m ≥ 1]. Es fácil
verificar que M ∈ latT y que TMM es de Banach de dimensión infinita.
(n)
Corolario 4. Sean X un espacio de Banach y T ∈ B(X). Si Tp
rador casi lleno para algún n, entonces T es casi lleno.
es un ope-
Demostración. Si T no es casi lleno, existen por el Lema 2 dos sucesiones:
+∞
∗
{fn }+∞
fk (T r xm ) = 0 para
n=1 ⊂ X y {xn }n=1 ⊂ X, tales que fk (xm ) = δkm , L
n
todo r, m, k ≥ 1. Para cada m ≥ 1 consideremos y2m−1 = j=1 δ1j xm , y2m =
94
Ln
j=1 δ2j xm y
Ln
h2m ( j=1 wj )
Ln
(n)
h2m−1 , h2m ∈ (Xp )∗ , tales que h2m−1 ( j=1 wn ) = fm (w1 ) y
Ln
(n)
= fm (w2 ), para todo j=1 wn ∈ Xp .
Corolario 5. Sean X1 , X2 espacios de Banach y T1 ∈ B(X1 ), T2 ∈ B(X2 )
operadores similares. Si T1 es casi lleno,entonces T2 es casi lleno.
Demostración. Si T2 no es casi lleno, existen fn ∈ X ∗ , xn ∈ X, tales que
fn (T2k xm ) = 0, ∀ n, m, k ≥ 1.
Sabemos que existe A ∈ B(X1 , X2 ) invertible, tal que A ◦ T1 ◦ A−1 = T2 . Se
deduce que A ◦ T1m ◦ A−1 = T2m (∗).
Sean hn = fn ◦ A, zn = A−1 xn . Es claro que hn (zm ) = δnm . Además por
(*)tenemos que hn (T1k zm ) = fn (T2k xm ) = 0, ∀ n, m, k ≥ 1, lo cual dice que T1
no es casi lleno.
El siguiente lema se obtuvo de las ideas expresadas en [3]:
Lema 3. Sea H un espacio de Hilbert separable y T ∈ B(H). Si
Tn
n
d
−→ 0
d
entonces existe una red de polinomios {hα }, tal que hα (T − I) −→ 0 y hα (X) =
Xqα (X) con qα (0) → 1.
2
Demostración.
LConsideremos N ∈ B(H) tal que N = 0, N 6= 0. Si el operador
A = (I + N ) T ∈ B(H (2) ), como (I + N )n = I + nN , entonces
M n
M
An
I
T
d
=
+N
−→ N
0.
n
n
n
L
L
Veamos L
que lat(N (T − I))(n) ⊂ lat(N 0)(n) . Consideremos cualquier
M ∈ lat(N (T − I))(n) y (x1 , y1 )..., (xn , yn ) ∈ M , luego
((N x1 , T y1 − y1 ), ..., (N xn , T yn − yn )) ∈ M.
Se deduce de aquı́ que (N x1 + x1 , T y1 )..., (N xn + xn , T yn )) ∈ M , por lo que
L (n)
L (n) m
L
M ∈ lat (N +I) T
. Como (N +I) T
= ((N +I) T )m )(n) =
L (n)
L
m
d
(Am )(n) , sigue que ( Am )(n) −→ (N
0)
y ası́ M ∈ lat(N
0). Existe por
L
L
d
lo tanto una red
{pα (X)} tal que pα (N (T − I)) −→ N
0.
Pnαde polinomios
k
α
α
Sea pα (X) = k=0 aα
k X , luego pα (N ) = a0 I + a1 N . Como
hpα N
M
(T − I) (x, y), (w, z)i =
α
aα
0 [hx, wi + hy, zi] + a1 hN x, wi + hpα (T − I)y, zi
M
→ h(N
0)(x, y), (w, z)i = hN x, wi (∗).
d
Tomando w = 0, se tiene hpα (T − I)y, zi → 0; es decir, pα (T − I) −→ 0.
95
2
Si x = w = N x0 6= 0 y z = 0, entonces aα
0 kN x0 k → 0 y pα (0) → 0. Sea
d
ahora hα (X) = pα (X) − pα (0), luego hα (T − I) −→ 0 y hα (X) = Xqα (X).
Tomando x = x0 , w = N x0 y sustituyendo estos valores en (*), se obtiene
2
2
α
que aα
1 kN x0 k → kN x0 k , por lo que qα (0) = a1 → 1.
Finalizamos presentando un resultado de Jaime Bravo en [3], el cual establece
un interesante contraste con el estudiado por nosotros en el Corolario 1.
Teorema 10. Sea H un espacio de Hilbert separable, T ∈ B(H) invertible. Si
T
n
d
A ∈ AlglatT {T }0 es tal que: An −→ 0, σ(T ) ⊂ π, y A es un operador lleno,
entonces
σp (A) es no vacı́o, o para todo M ∈ latT, M 6= (0), tenemos que
T∞
n
T
M
6= (0).
n=0
T∞
Demostración. Si existe M0 ∈ latT tal que n=0 T n M0 = (0), entonces claramente el operador T |M0 no es lleno. Por lo tanto podemos hallar un M ∈ latT ,
tal que M ⊂ M0 y dim T n M ΘT n+1 M = 1, ∀n ≥ 1 Sea {en }∞
n=0 una familia ortonormal con en ∈ T n M T n+1 M . Veamos que {en }∞
n=0 es una base ortonormal de M . En efecto si x ∈ M con hx, en i = 0, ∀ n ≥ 0, como
x = αen + w, w ∈ T M ⇒ hx, en i = α ⇒ x ∈ T M . Por un proceso inductivo
llegamos a que x ∈ T n M, ∀ n ≥ 0 ⇒ x = 0.
Tenemos que
T en = αn en+1 + yn+2 (yn+2 ∈ T n+2 M )
y Aen = βen + zn+1 (zn+1 ∈ T n+1 M ),
(∗)
(∗∗)
ya que A ∈ AlglatT . Como en = αn T −1 en+1 + T −1 yn+2 (T −1 yn+2 ∈ T n+1 M ),
sigue que an 6= 0, y usando el hecho de que AT en = T Aen se tiene βn αn =
βn+1 αn , por tanto βn = βn+1 = β, ∀ n ≥ 0.
Notemos que h(A−βI)n e0 , e0 i = 0 ∀n ≥ 1 ⇒ A−βI no es un operador lleno.
Si |β| > 1, luego β ∈ ρ∞ (A). Sea N ∈ lat(A − βI). Tenemos que N ∈ latA y por
Sarason N ∈ lat(A − βI)−1 . Se deduce que (A − βI)−1 (N ) ⊂ N =⇒ N ⊂ (A −
βI)N ⊂ N =⇒ (A − βI)N = N , lo cual dice que A − βI es lleno y por lo tanto
obtenemos una contradicción. Sea |β| < 1, luego β y 0 pertenecen a una misma
componente conexa de ρ∞ (A). Si N ∈ lat(A − βI), luego N ∈ latA = latA−1
(por ser A lleno e invertible). Por Sarason tenemos que N ∈ lat(A − βI)−1 y
por el mismo razonamiento anterior llegamos a que A − βI es un operador lleno,
lo que es contadictorio. Se deduce que |β| = 1.
Sin pérdida de generalidad podemos suponer que β = 1. Vamos a demostrar
que (A−I)en = 0, ∀ n ≥ 0. Veamos que M (A−I)n M ⊂ M (A−I)M, ∀n ≥ 2.
En efecto, sea x ∈ M (A − I)n M . Si y = (A − I)z y {hα (X)} es una red
de polinomios como la construida en el lema anterior, luego
hhα ((A − I))z, xi = hqα (A − I)y, xi = qα (0)hy, xi → 0,
96
de donde hy, xi = 0, y ası́ x ∈ M (A − I)M .
De lo anterior deducimos que ((A − I)M ⊂ (A − I)n M para todo n ≥ 2.
Finalmente observemos que de (**), se deduce que (A − I)M ⊂ T M , y por
un proceso inductivoTllegamos a que (A − I)n M ⊂ T n M para todo n ≥ 2, de
+∞
donde (A − I)M ⊂ { n=0 T n M = {0}, por tanto (A − I)M = 0.
Referencias
[1] Bachman G. and Narici L. Functional Analysis. Academic Press. New York
and London (1966)
[2] Bravo J. Relations between latT, latT −1 , latT 2 and operators with compact imaginary parts. Ph. D. Dissertation. University of California Berkeley
(1980).
T
0
[3] Bravo J. Operadores regulares en AlglatT {T } . Departamento de Matemáticas. Facultad Experimental de Ciencias, LUZ (2007).
[4] Karanasios S. Full operators and approximation of inverses. J. London
Math. Soc. 36, 295-304. (1982).
[5] Karanasios S. Full operators on reflexive Banach spaces. Bull. Greek Math.
Soc. 36, 81-86. (1994).
[6] Kubrusly C. Hilbert Space Operators. A Problem Solving Approach. Birkhauser, Boston. (2008).
[7] Rosales E. Operadores casi llenos y de radio numérico alcanzable. Boletı́n
de la Asociación Matemática Venezolana. Vol. XIX, No. 2. 147-154. (2012).
[8] Vieira P. C. M, Malenbranche H. and Kubrusly C. S. On uniform stability.
Advances in Mathematical Sciences and Applications. 8, 95-100. (1998).
Rosales Edixo, Klarys Fereira
Universidad del Zulia, Facultad Experimental de Ciencias
Departamento de Matemáticas. Maracaibo, Venezuela
[email protected] [email protected]
97
Coefficient bounds for subclasses of bi-univalent
functions defined by the Sălăgean derivative
operator
K. Vijaya, M. Kasthuri and G. Murugusundaramoorthy
Abstract. In this paper, we introduce and investigate a new subclass of the function class Σ of bi-univalent functions defined in the
open unit disk, which are associated with the Sălăgean operator and
satisfy some subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients |a2 | and |a3 | for functions
in the new subclass introduced here. Several (known or new) consequences of the results are also pointed out. Further making use of
the values of a22 and a3 , we obtain the Fekete-Szegö result for the
function class MkΣ (λ, φ).
2010 Mathematics Subject Classification: 30C45.
Keywords and Phrases: Univalent, bi-univalent, starlike and convex functions,
coefficient bounds.
Resumen. En este artı́culo introducimos e investigamos una nueva
subclase de la clase Σ de funciones bi-univalentes definidas en el disco
abierto unitario, las cuales están asociadas al operador Sălăgean y
satisfacen algunas condiciones de subordinación. Obtenemos además
estimaciones para los coeficientes |a2 | y |a3 | de Taylor-Maclaurin
para funciones en la nueva subclase. Varias (conocidas o nuevas)
consecuencias de los resultados son también señaladas. Haciendo
uso de los valores de a22 y a3 deducimos el resultado de Fekete-Szegö
para la clase MkΣ (λ, φ).
1
Introduction
Let A denote the class of analytic functions of the form
f (z) = z +
∞
X
n=2
an z n
(1.1)
98
normalized by the conditions f (0) = 0 = f 0 (0) − 1 defined in the open unit disk
4 = {z ∈ C : |z| < 1}.
Let S be the subclass of A consisting of functions of the form (1.1) which are
also univalent in 4. Let S ∗ (α) and K(α) denote the subclasses of S, consisting
of starlike and convex functions of order α, 0 ≤ α < 1, respectively.
The Koebe one quarter theorem [4] ensures that the image of 4 under
every univalent function f ∈ A contains a disk of radius 41 . Thus every univalent function f has an inverse f −1 satisfying f −1 (f (z)) = z, (z ∈ 4) and
f (f −1 (w)) = w (|w| < r0 (f ), r0 (f ) ≥ 41 ). A function f ∈ A is said to be biunivalent in 4 if both f and f −1 are univalent in 4. Let Σ P
denote the class of
bi-univalent functions defined in the unit disk 4. Since f ∈
has the Maclaurian series given by (1.1), a computation shows that its inverse g = f −1 has the
expansion
g(w) = f −1 (w) = w − a2 w2 + (2a22 − a3 )w3 + · · ·
(1.2)
An analytic function f is subordinate to an analytic function g, written
f (z) ≺ g(z), provided there is an analytic function w defined on 4 with w(0) = 0
and |w(z)| < 1 satisfying f (z) = g(w(z)). Ma and Minda [8] unified various
subclasses of starlike and convex functions for which either of the quantities
z f 0 (z)
f (z)
or
1+
z f 00 (z)
f 0 (z)
is subordinate to a more general superordinate function. For this purpose,
they considered an analytic function ϕ with positive real part in the unit disk
4, ϕ(0) = 1, ϕ0 (0) > 0 and ϕ maps 4 onto a region starlike with respect to
1 and symmetric with respect to the real axis. The class of Ma-Minda starlike
0
(z)
functions consists of functions f ∈ A satisfying the subordination z ff(z)
≺ ϕ(z).
Similarly, the class of Ma-Minda convex functions consists of functions f ∈ A
00
(z)
satisfying the subordination 1+ z ff0 (z)
≺ ϕ(z). A function f is bi-starlike of MaMinda type or bi-convex of Ma-Minda type if both f and f −1 are respectively
Ma-Minda starlike or convex. These classes are denoted respectively by SΣ∗ (ϕ)
and KΣ (ϕ). In the sequel, it is assumed that ϕ is an analytic function with
positive real part in the unit disk 4, satisfying ϕ(0) = 1, ϕ0 (0) > 0 and ϕ(4)
is symmetric with respect to the real axis. Such a function has a series expansion
of the form
ϕ(z) = 1 + B1 z + B2 z 2 + B3 z 3 + · · · ,
(B1 > 0).
(1.3)
Several authors have introduced and investigated subclasses of bi-univalent
functions and obtained bounds for the initial coefficients (see [1, 2, 5, 7, 6, 13,
15, 17]). In this paper, making use of the Sălăgean [12] differential operator
Dk : A → A
Coefficient bounds for subclasses
99
defined by
D0 f (z)
=
f (z),
D f (z)
= Df (z) = zf 0 (z),
Dk f (z)
= D(Dk−1 f (z)) = z(Dk−1 f (z))0 , k ∈ N = {1, 2, 3, . . .}
∞
X
= z+
nk an z n , k ∈ N0 = N ∪ {0},
(1.4)
1
Dk f (z)
n=2
further for functions g of the form (1.2) we define
Dk g(w) = w − a2 2k w2 + (2a22 − a3 )3k w3 + · · ·
(1.5)
and introduce two new subclass of bi-univalent functions to obtain the estimates
on the coefficients |a2 | and |a3 | by subordination.
2
Bi-Univalent function class MkΣ (λ, φ)
In this section, we introduce a subclass MkΣ (λ, φ) of Σ and find estimate on the
coefficients |a2 | and |a3 | for the functions in this new subclass, by subordination.
Definition 2.1. For 0 ≤ λ ≤ 1, a function f ∈ Σ of the form (1.1) is said to
be in the class MkΣ (λ, φ) if the following subordination holds:
Dk+2 f (z)
Dk+1 f (z)
+ λ k+1
≺ φ(z)
k
D f (z)
D
f (z)
(2.1)
Dk+1 g(w)
Dk+2 g(w)
+
λ
≺ φ(w),
Dk g(w)
Dk+1 g(w)
(2.2)
(1 − λ)
and
(1 − λ)
where z, w ∈ ∆ and g is given by (1.2).
Remark 2.2. Suppose f ∈ Σ. If λ = 0, then MkΣ (λ, φ) ≡ SΣk (φ): thus f ∈ SΣk (φ)
if the following subordination holds:
Dk+1 f (z)
Dk+1 g(w)
≺ φ(z) and
≺ φ(w),
k
D f (z)
Dk g(w)
k
k
Remark 2.3. Suppose f ∈ Σ. If λ = 1, then MkΣ (λ, φ) ≡ KΣ
(φ) : thus f ∈ KΣ
(φ)
if the following subordination holds:
Dk+2 g(w)
Dk+2 f (z)
≺
φ(z)
and
≺ φ(w),
Dk+1 f (z)
Dk+1 g(w)
100
It is of interest to note that M0Σ (0, φ) = SΣ∗ (φ), M0Σ (1, φ) = KΣ (φ).
In order to prove our main results, we require the following Lemma:
Lemma 2.4. [10] If a function p ∈ P is given by
p(z) = 1 + p1 z + p2 z 2 + · · ·
(z ∈ ∆),
then
|pi | 5 2
(i ∈ N),
where P is the family of all functions p, analytic in ∆, for which
p(0) = 1
and
< p(z) > 0
(z ∈ ∆).
Theorem 2.5. Let f given by (1.1) be in the class MkΣ (λ, φ). Then
√
B1 B1
|a2 | ≤ p
|[2(1 + 2λ)3k − (1 + 3λ)22k ]B12 + (1 + λ)2 22k (B1 − B2 )|
and
|a3 | ≤
B1
+
2(1 + 2λ)3k
B1
2k (1 + λ)
(2.3)
2
,
(2.4)
where 0 ≤ λ ≤ 1.
Proof. Let f ∈ MkΣ (λ, φ) and g = f −1 . Then there are analytic functions
u, v : 4 −→ 4, with u(0) = 0 = v(0), satisfying
Dk+1 f (z)
Dk+2 f (z)
+ λ k+1
= ϕ(u(z))
k
D f (z)
D
f (z)
(2.5)
Dk+2 g(w)
Dk+1 g(w)
+ λ k+1
= ϕ(v(w)).
k
D g(w)
D
g(w)
(2.6)
(1 − λ)
and
(1 − λ)
Define the functions p(z) and q(z) by
p(z) :=
1 + u(z)
= 1 + p1 z + p2 z 2 + · · ·
1 − u(z)
q(z) :=
1 + v(z)
= 1 + q1 z + q2 z 2 + · · ·
1 − v(z)
and
or, equivalently,
u(z) :=
p(z) − 1
1
p2
=
p1 z + p2 − 1 z 2 + · · ·
p(z) + 1
2
2
(2.7)
and
v(z) :=
q(z) − 1
1
q2
=
q1 z + q2 − 1 z 2 + · · · .
q(z) + 1
2
2
101
(2.8)
Then p(z) and q(z) are analytic in 4 with p(0) = 1 = q(0). Since u, v : 4 → 4,
the functions p(z) and q(z) have a positive real part in 4, |pi | ≤ 2 and |qi | ≤ 2.
Using (2.7) and (2.8) in (2.5) and (2.6) respectively, we have
Dk+2 f (z)
1
p2
Dk+1 f (z)
+ λ k+1
=ϕ
p1 z + p2 − 1 z 2 + · · ·
(2.9)
(1 − λ) k
D f (z)
D
f (z)
2
2
and
(1−λ)
Dk+1 g(w)
Dk+2 g(w)
+λ k+1
=ϕ
k
D g(w)
D
g(w)
1
q2
q1 w + q2 − 1 w2 + · · ·
. (2.10)
2
2
In light of (1.1) - (1.5), and from (2.9) and (2.10), we have
1 + (1 + λ)2k a2 z + [2(1 + 2λ)3k a3 − (1 + 3λ)22k a22 ]z 2 + · · ·
1
p21
1
1
2
= 1 + B1 p1 z + B1 (p2 − ) + B2 p1 z 2 + · · ·
2
2
2
4
and
1 − (1 + λ)2k a2 w + {[(8λ + 4)3k − (3λ + 1)22k )]a22 − 2(1 + 2λ)3k a3 }w2 + · · ·
1
1
q2
1
= 1 + B1 q1 w + B1 (q2 − 1 ) + B2 q12 w2 + · · ·
2
2
2
4
which yields the following relations:
(1 + λ)2k a2
=
−(1 + 3λ)22k a22 + 2(1 + 2λ)3k a3
=
−(1 + λ)2k a2
=
1
B1 p 1
2
p2
1
1
B1 (p2 − 1 ) + B2 p21
2
2
4
1
B1 q 1
2
(2.11)
(2.12)
(2.13)
and
(4(1 + 2λ)3k − (1 + 3λ)22k )a22 − 2(1 + 2λ)3k a3 =
1
q2
1
B1 (q2 − 1 ) + B2 q12 . (2.14)
2
2
4
From (2.11) and (2.13) it follows that
p1 = −q1
(2.15)
102
and
8(1 + λ)2 22k a22 = B12 (p21 + q12 ).
(2.16)
From (2.12), (2.14) and (2.16), we obtain
a22 =
B13 (p2 + q2 )
.
4[{2(1 + 2λ)3k − (1 + 3λ)2k }B12 + (1 + λ)2 22k (B1 − B2 )]
Applying Lemma 2.4 to the coefficients p2 and q2 , we have
√
B1 B1
|a2 | ≤ p
|[2(1 + 2λ)3k − (1 + 3λ)22k ]B12 + (1 + λ)2 22k (B1 − B2 )|
(2.17)
(2.18)
By subtracting (2.14) from (2.12) and using (2.15) and (2.16), we get
a3 =
B12 (p21 + q12 )
B1 (p2 − q2 )
+
.
8(1 + λ)2 22k
8(1 + 2λ)3k
Applying Lemma 2.4 once again to the coefficients p1 , p2 , q1 and q2 , we get
2
B1
B1
|a3 | ≤
+
.
(2.19)
2(1 + 2λ)3k
2k (1 + λ)
So the proof is complete.
For k = 0, Theorem 2.5 yields the following corollary.
Corollary 2.6. Let f given by (1.1) be in the class MΣ (λ, φ). Then
√
B1 B1
|a2 | ≤ p
|(1 + λ)B12 + (1 + λ)2 (B1 − B2 )|
and
|a3 | ≤
B1
+
2(1 + 2λ)
B1
1+λ
(2.20)
2
.
(2.21)
For the class of strongly starlike functions of order α(0 < α ≤ 1), the function
φ is given by
α
1+z
φ(z) =
= 1 + 2αz + 2α2 z 2 + · · · (0 < α ≤ 1),
(2.22)
1−z
which gives B1 = 2α and B2 = 2α2 . On the other hand, for the class of starlike
functions of order β(0 ≤ β < 1), if we take
φ(z) =
1 + (1 − 2β)z
= 1 + 2(1 − β)z + 2(1 − β)z 2 + · · ·
1−z
then we get B1 = B2 = 2(1 − β).
(0 ≤ β < 1), (2.23)
Remark 2.7. If f ∈ MkΣ (λ,
1+z
1−z
α
103
) then, we have the following estimates for
the coefficients |a2 | and |a3 | :
2α
|a2 | ≤ p
k
|{2(1 + 2λ)3 − (1 + 3λ)22k }α + (1 − α)(1 + λ)2 22k |
and
|a3 | ≤
4α2
α
+
.
(1 + λ)2 22k
(1 + 2λ)3k
For functions f ∈ MkΣ (λ, 1+(1−2β)z
), the inequalities (2.3) and (2.4) yields the
1−z
following estimates
s
2(1 − β)
|a2 | ≤
|2(1 + 2λ)3k − (1 + 3λ)22k |
and
|a3 | ≤
4(1 − β)2
(1 − β)
+
.
(1 + λ)2 22k
(1 + 2λ)3k
Remark 2.8. We note that, for k = 0, these estimates coincides with the results
stated in [17].Consequently, when λ = 0 and λ = 1 one has the estimates for
the classes SΣ∗ (α), SΣ∗ (β)(see[13]) and KΣ (α), KΣ (β) (see [2, 14]) respectively.
Due to Frasin and Aouf [3] and Panigarhi and Murugusundaramoorthy [11]
we define the following new subclass involving the Sălăgean operator [12].
3
Bi-Univalent function class FΣk (µ, φ)
Definition 3.1. For 0 ≤ µ ≤ 1, a function f ∈ Σ of the form (1.1) is said to
be in the class FΣk (µ, φ) if the following subordination holds:
(1 − µ)
Dk f (z)
+ µ(Dk f (z))0 ≺ φ(z)
z
(3.1)
and
Dk g(w)
+ µ(Dk g(w))0 ≺ φ(w)
w
where z, w ∈ ∆, g is given by (1.2) and Dk f (z) is given by (1.4).
(1 − µ)
(3.2)
k
Remark 3.2. Suppose f (z) ∈ Σ. If µ = 0, then FΣk (0, φ) ≡ HΣ
(φ): thus, f
k
belongs to HΣ (φ) if the following subordination holds:
Dk f (z)
Dk g(w)
≺ φ(z) and
≺ φ(w),
z
w
104
Remark 3.3. Suppose f (z) ∈ Σ. If µ = 1, then FΣk (1, φ) ≡ PΣk (φ): thus, f ∈
PΣk (φ) if the following subordination holds:
(Dk f (z))0 ≺ φ(z) and (Dk g(w))0 ≺ φ(w),
It is of interest to note that FΣ0 (µ, φ) = FΣ∗ (µ, φ).
Theorem 3.4. Let f given by (1.1) be in the class FΣk (µ, φ). Then
√
B1 B1
|a2 | ≤ p
|(1 + 2µ)3k B12 + (1 + µ)2 22k (B1 − B2 )|
(3.3)
and
|a3 | ≤ B1
B1
1
+
(1 + µ)2 22k
(1 + 2µ)3k
.
(3.4)
Proof. Proceeding as in the proof of Theorem 2.5 we can arrive the following
relations.
(1 + µ)2k a2
=
(1 + 2µ)3k a3
=
−(1 + µ)2k a2
=
1
B1 p1
2
1
p2
1
B1 (p2 − 1 ) + B2 p21
2
2
4
1
B1 q1
2
and
2(1 + 2µ)3k a22 − (1 + 2µ)3k a3 =
1
q2
1
B1 (q2 − 1 ) + B2 q12 .
2
2
4
(3.5)
(3.6)
(3.7)
(3.8)
From (3.5) and (3.7) it follows that
p1 = −q1
(3.9)
8(1 + µ)2 22k a22 = B12 (p21 + q12 ).
(3.10)
and
From (3.6), (3.8) and (3.10), we obtain
a22 =
4[(1 +
B13 (p2 + q2 )
.
+ (1 + µ)2 22k (B1 − B2 )]
2µ)3k B12
Applying Lemma 2.4 to the coefficients p2 and q2 , we immediately get the
desired estimate on |a2 | as asserted in (3.3).
105
By subtracting (3.8) from (3.6) and using (3.9) and (3.10), we get
a3 =
B12 (p21 + q12 )
B1 (p2 − q2 )
+
.
2
2k
8(1 + µ) 2
4(1 + 2µ)3k
Applying Lemma 2.4 to the coefficients p1 , p2 , q1 and q2 , we get the desired
estimate on |a3 | as asserted in (3.4).
Making use of the values of a22 and a3 , and motivated by the recent work of
Zaprawa [19] we prove the following Fekete-Szegö result for the function class
MkΣ (λ, φ).
Theorem 3.5. Let the function f (z) given by (1.1) be in the class MkΣ (λ, φ)
and η ∈ R. Then
|a3 −ηa22 | ≤
(1 + λ)2 (B1 − B2 )22k + (1 + 3λ)B12 2k
B1
,
if
|1−η|
≤
1+
2(1 + 2λ)3k
2(1 + 2λ)B12 3k
and
|a3 − ηa22 | ≤
|[2(1 +
whenever |1 − η| ≥ 1 +
2λ)3k
B13 |(1 − η)|
,
− (1 + 3λ)2k ]B12 + (1 + λ)2 (B1 − B2 )22k |
(1 + λ)2 (B1 − B2 )22k + (1 + 3λ)B12 2k
.
2(1 + 2λ)B12 3k
Proof. From (2.12) and (2.14) and by using (2.15), we get
a3 = a22 +
B1 (p2 − q2 )
.
8(1 + 2λ)3k
By simple calculation using (2.17), we get
1
1
2
p2 + Θ(η) −
q2 ,
a3 − ηa2 = B1 Θ(η) +
8(1 + 2λ)3k
8(1 + 2λ)3k
where
Θ(η) =
4[2(1 +
2λ)3k
B12 (1 − η)
.
− (1 + 3λ)2k ]B12 + 4(1 + λ)2 (B1 − B2 )22k
Since all Bj are real and B1 > 0, we have

B1
1

0 ≤ |Θ(η)| < 8(1+2λ)3
k,
 2(1+2λ)3k ,
2
|a3 − ηa2 | ≤

1
 4B1 |Θ(η)|,
|Θ(η)| ≥ 8(1+2λ)3
k,
which completes the proof.
106
Remark 3.6. We note that, for k = 0 these estimates coincides with the results
stated in [3]. Consequently, when λ = 0 and λ = 1 one has the estimates for
∗
∗
the classes HΣ
(α), HΣ
(β) (see[13]) and Deniz[5]. Putting λ = 0 (resp. λ = 1)
in Theorem 3.5, we can state the Fekete-Szegö inequality for the function class
k
SΣk (φ) (resp. KΣ
(φ)).
Acknowledgement: The authors thank the referee for his insightful suggestions to improve this paper in the present form.
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K. Vijaya, M. Kasthuri and G. Murugusundaramoorthy∗
School of Advanced Sciences, VIT University
Vellore - 632014, India.
[email protected], [email protected], [email protected]
∗ Corresponding author
Boletı́n de la Asociación Matemática Venezolana, Vol. XXI, No. 2 (2014) 109
On multipliers of some new analytic function
spaces of BMOA type in the polydisc
R. Shamoyan and S. Kurilenko
Abstract. We define two new scales of BMOA type analytic function spaces in the polydisc. We provide several new results concerning coefficient multipliers of these two new BMOA analytic function
spaces in the polydisc. Our results extend previously known assertions.
2010 Mathematics Subject Classification: Primary 42B15, Secondary 42B30.
Keywords and Phrases: analytic functions, multipliers, polydisc theorems.
Resumen. Definimos dos nuevas escalas de espacios de funciones
analı́ticas del tipo BMOA en el polidisco. Mostramos varios resultados nuevos referentes a coeficientes multiplicadores de esas dos
nuevos espacios de funciones analı́ticas BMOA en el polidisco. Nuestros resultados extienden afirmaciones ya conocidas.
1
Introduction
The goal of this paper is to continue the investigation of spaces of coefficient
multipliers of some new holomorphic scales of functions in higher dimension,
namely in the unit polydisc. This note is a continuation of a series of recent
notes on this topic [1], [15], [16], [19]. The spaces we study in this note are
natural extensions of the classical BMOA-type spaces in the unit polydisc. Note
that pointwise and coefficient multipliers of analytic function spaces of Hardy
and Bergman type in one and higher dimension were studied intensively by
many authors during past several decades, see for example [3], [18], [5] and
various references there. The investigation of coefficient multipliers of analytic
mixed norm function spaces in the unit polydisc was started in particular in
recent papers of the first author, see [15], [16], [17], [19]. Below we list standard
This work was supported by the Russian Foundation for Basic Research (Grant 13-3530197508)
110
notation and definitions which are needed and in the next section we formulate
the main results of this note. To be more precise we consider two new scales of
analytic functions of BMOA type in the unit polydisc. One of them is the dual
of analytic Hardy class H 1 in the unit polydisc and the second one is based on
well-known characterization of BMOA spaces via Carleson-type measure in the
unit disk. Both scales in one dimension obviously concede with the classical
BMOA in the unit disk. Note we do not provide sharp results in this paper,
but our results are far reaching extensions of onedimensional known theorems
on coefficient multipliers of BMOA type spaces (see [3], [7], [10], [11]). As in
many previous papers on this topic in this paper we concentrate in searching of
conditions on coefficient multipliers of BMOA type space in the unit polydisc
via conditions on Mp (f, r) type functions (see definitions below). Let U =
{z ∈ C : |z| < 1} be the unit disc in C, T = ∂U = {z ∈ C : |z| = 1}, U n
is the unit polydisc in Cn , Tn ⊂ ∂U n is the distinguished boundary of U n ,
Z+ = {n ∈ Z : n ≥ 0}, Zn+ is the set of all multi indexes and I = [0, 1).
We use the following notation: for z = (z1 , . . . , zn ) ∈ Cn and k = (k1 , . . . , kn )
in Zn+ we set z k = z1k1 · · · znkn and for z = (z1 , . . . , zn ) ∈ U n and γ ∈ R we set
(1 − |z|)γ = (1 − |z1 |)γ · · · (1 − |zn |)γ and (1 − z)γ = (1 − z1 )γ · · · (1 − zn )γ . Next,
for z ∈ Rn and w ∈ Cn we set wz = (w1 z1 , . . . , wn zn ). Also, for k ∈ Zn+ and
a ∈ R we set k + a = (k1 + a, . . . , kn + a). For z = (z1 , . . . , zn ) ∈ Cn we set
z = (z 1 , . . . , z n ). For k ∈ (0, +∞)n we set Γ(k) = Γ(k1 ) · · · Γ(kn ).
The Lebesgue measure on Cn ∼
= R2n is denoted by dV (z), the normalized
n
Lebesgue measure on T is denoted by dξ = dξ1 . . . dξn and dR = dR1 . . . dRn
is the Lebesgue measure on [0, +∞)n .
n
n
The space of all holomorphic functions
P in U iskdenoted by H(U ). Every
n
f ∈ H(U ) admits an expansion f (z) = k∈Zn ak z . For β > −1 the operator
+
of fractional differentiation is defined by (see [8])
Dβ f (z) =
Γ(k + β + 1)
ak z k ,
Γ(β + 1)Γ(k + 1)
X
k∈Zn
+
z ∈ U n.
(1)
Similarly we can easily define the operator of fractional derivative for βvector, β = (β1 , . . . , βn ), βj > −1, j = 1, . . . , n.
~
Dβ f (z) =
Γ(kj + βj + 1)
ak ...k z k1 . . . znkn
Γ(βj + 1)Γ(kj + 1) 1 n 1
X
k∈Zn
+
For f ∈ H(Un ), 0 < p < ∞ and r ∈ I n we set
Z
Mp (f, r) =
Tn
|f (rξ)|p dξ
1/p
, dξ = dξ1 . . . dξn
(2)
On multipliers of some new analytic function spaces
111
with the usual modification to include the case p = ∞. For 0 < p ≤ ∞ we
define standard analytic Hardy classes in the polydisc:
H p (Un ) = {f ∈ H(U n ) : kf kH p = sup Mp (f, r) < ∞}.
(3)
r∈I n
For n = 1 these spaces are well studied. The topic of multipliers of Hardy
spaces in the polydisc is relatively new, see, for example, [7], [3], [18], [15], [16].
These spaces are Banach spaces for p > 1 and complete metric spaces for all
other positive values of p. Also, for 0 < p ≤ ∞, 0 < q < ∞ and α > 0 we have
mixed (quasi) norm spaces, defined below as follows
n
Ap,q
α (U ) =
f ∈ H(U n ) : kf kqAp,q
=
α
Z
In
Mpq (f, R)(1 − R)αq−1 dR < ∞ . (4)
If we replace the integration by I n above by integration by unit interval I
then we get other similar to these Ap,q
α analytic spaces but on subframe and we
denote them by Bαp,q . These spaces are Banach spaces for cases when both p and
q are bigger than one, and they are complete metric spaces for all other values
of parameters. We refer the reader for these classes in the unit ball and the
unit disk to [3], [12], [13], [14] and references therein. Multipliers between Ap,q
α
spaces on the unit disc were studied in detail in [3], [7], [10], [11]. Multipliers
of spaces on subframe are less studied. In forthcoming paper we will study
multipliers of related B p,q spaces. As we see from assertions below spaces of
multipliers of classes on subframe depend from dimension n.
We denote positive constants by c (or C), sometimes we indicate dependence
of a constant on a parameter by using a subscript, for example, Cq .
We define now one of the main objects of this paper.
For 0 < p, q < ∞ and α > 0 we consider Lizorkin-Triebel spaces Fαp,q (U n ) =
p,q
Fα consisting of all f ∈ H(U n ) such that
kf kpFαp,q =
Z
Tn
Z
|f (Rξ)|q (1 − R)αq−1 dR
p/q
dξ < ∞.
(5)
In
It is not difficult to check that those spaces are complete metric spaces, if
min(p, q) ≥ 1 they are Banach spaces. If we simply replace the integration by
I n above by integration by unit interval I then we get other similar to these
Fαp,q analytic spaces, but on subframe and we denote them by Tαp,q . They were
studied in [20]. We refer the reader to this paper for various other properties
on these type classes. We provide some new assertions on multipliers of this
new Fαp,q type spaces on subframe below. We note that this F p,q scale of spaces
p,p
(see [8] for a detailed
includes, for p = q, weighted Bergman spaces Apα = F α+1
p
account of these spaces). On the other hand, for q = 2 these spaces coincide with
112
p,2
, for this well known fact see [16], [8]
Hardy-Sobolev spaces namely Hαp = F α+1
2
and references therein.
Finally, for α ≥ 0 and β ≥ 0 we set
n
n
α
β
∞,∞
A∞,∞
α,β (U ) = {f ∈ H(U ) : kf kAα,β = sup (M∞ (D f, r))(1 − r) < ∞}. (6)
r∈I n
This is a Banach space and it is related with the well-known Bloch class in the
polydisc (see [3]). The Bloch class studied by many authors in various papers
(see, for example, [3], [6] for unit disk case and for the polydisc case and also
various references there). The Bloch space is a same
n
A∞,∞
α,β (U )
space with α = β = 1 in definition above. This space is also a Banach space.
For all positive values of p and s we introduce the following two new spaces. We
note first replacing q by ∞ in a usual way we will arrive at some other spaces
F p,∞ (the limit case of Fαp,q classes). The limit space case F p,∞,s (U n ) is defined
as a space of all analytic functions f in the polydisc such that the function
φ(ξ) = supr∈I n |f (rξ)|(1 − r)s , ξ ∈ Tn is in Lp (Tn , dξ), s ≥ 0, 0 < p ≤ ∞.
Finally, the limit space case Ap,∞,s (U n ) is the space of all analytic functions
f in the polydisc such that (see [3])
sup Mp (f, r)(1 − r)s < ∞, s ≥ 0, 0 < p ≤ ∞.
r∈I n
Obviously the limit F p,∞,s space is embedded in the last space we defined.
These last two spaces are Banach spaces for all p ≥ 1 and they are complete
metric spaces for all other positive values of p.
The following definition of coefficient multipliers is well known in the unit
disk. We provide a natural extension to the polydisc setup. This definition can
be seen in papers of the first author before.
Definition 1. Let X and Y be quasi normed subspaces of H(U n ). A sequence
c = {ck }k∈Zn+ is said to be a coefficient multiplier from X to Y if for any
P
function f (z) = k∈Zn ak z k in X the function h = Mc f defined by h(z) =
+
P
ck ak z k is in Y . The set of all multipliers from X to Y is denoted by
k∈Zn
+
MT (X, Y ).
The problem of characterizing the space of multipliers (pointwise multipliers
and coefficient multipliers) between various spaces of analytic functions is a
classical problem in complex function theory, there is vast literature on this
subject (see [7], [8], [10], [11], and references therein).
In this paper we are looking for some extensions of already known classical
theorems, namely we are interested in spaces of multipliers acting into analytic
113
BMOA type spaces in the unit polydisc and from these spaces into certain well
studied classes like mixed norm spaces, Bergman spaces and Hardy spaces. We
note that the analogue of this problem of description of spaces of multipliers
in Rn for various functional spaces in Rn was considered previously by various
authors in recent decades (see [3]). The intention of this note is to study the
spaces of coefficient multipliers of new analytic function spaces in polydisc.
As we indicated this topic is well-known and various results by many authors
were published in recent decades in this area starting from classical papers
of Hardy and Littlewood (see for example [3]) for many known results and
various references there. Nevertheless the study of coefficient multipliers of
analytic function spaces in higher dimension namely in the polydisc is a new
area of research and only several sharp results are known till now (see, for
example, [3], [1], [15] – [18] and various references there). We complement these
results and our results from [15], [16], [17] in this note providing new theorems
on coefficient multipliers of some new analytic function spaces in the polydisc.
We also define by dm2 (ξ) the Lebesgue measure on the unit disk and replacing 2 by 2n the Lebesgue measure on the polydisc.
Let Lp~ (~
α) be the space of all measurable functions in U n so that
Z
kf kLp~ (~α) =
(1−|ξn |)αn
U
Z
(1−|ξn−1 |)αn−1 ...
U
! pp2
1
×dm2 (ξ1 )
Z
(1−|ξ1 |)α1 |f (ξ1 , ..., ξn )|p1 ×
U
! p pn
n−1
...dm2 (ξn−1 )
! p1
n
dm2 (ξn )
< +∞.
→
− −
where all pj are positive and all αj > −1 for any j = 1, ..., n Let H p (→
α) =
p
~
n
L (~
α) ∩ H(U ). These spaces are one of the main objects of study of this paper
(see [1], [2] for properties of these new classes).
We define now new mixed norm weighted Hardy analytic spaces in the polydisc in the following natural way.
Let
! pp2
! pp3
! p1 n
Z
Z Z
n Y
1
2
→
−
→
−
p
p1
=
sup
H→
...
|f
(r
ξ
)|
dξ
(1 − rj )αj ;
dξ
.
.
.
dξ
−
1
2
n
α
rj <1
T
T
T
j=1
pj ∈ (0; ∞]; αj ≥ 0, j = 1, . . . , n. Note in unit disk we get well-studied classical
weighted Hardy spaces [3], [8], [9], [10], [19], H0p,...,p = H p (U n ). The main
objects of this paper are two new BMOA type spaces in higher dimension.
BMOA type spaces in higher dimension can be defined differently starting from
their various onedimensional characteristics (see [3] and references there). We
first define BMOA in the polydisc as dual space of classical analytic Hardy
114
space H 1 in the polydisc which we mentioned above. We note this serves as
direct extension of BMOA spaces from the unit disk (see [3]). Another natural
way to extend directly BMOA spaces from unit disk to higher dimension is to
use their characterizations in unit disk via Carleson type measures, this type
approach was used already by first author before. Namely we define the other
q,∞
be a space of analytic functions
main object of this paper as follows. Let Fs,α
in the unit polydisc, so that
Z
Un
|f (z)|q (1 − |z|)s (1 − |w|)α
dm2n (z)
|1 − wz|α+1
is a bounded function in the unit polydisc as a function of w variable. We assume
that s > −1 and other values of parameters above are positive numbers. Note
both spaces coincide in the unit disk with each other and with classical BMOA
space (see [3]). These spaces are the main objects of this paper.
2
Preliminary results
For formulation of all our main results of this note we need several lemmas,
almost all of them are known and taken from recently published papers. To
prove our results we need to use a simple idea. We connect our BMOA type
analytic function spaces in the polydisc with much simpler well-studied spaces
in the polydisc and using these connections (embeddings) we find information
about sizes of spaces of coefficient multipliers we search for. Another important
ingredients of our work are estimates of Bergman kernel in the polydisc in some
new spaces with usual and unusual quazinorms in the polydisc. Note much
information concerning this was taken from recent papers of the first author on
this topic (see [20] and references there). We mention also [21] where some new
results also can be found. As usual we define positive constants by C (or c)
with various (mainly lower) indexes depending on situation.
In particular we need the following lemma 1:
Lemma 1 (see [1], [2]). Let g(z1 , ..., zn ) =
n
Q
j=1
1
(1−zj )βj
, gr (z) = g(rz), r ∈
I n , zj ∈ U, βj > 0, j = 1, ..., n, gr (z) = g(rz). Then for some large enough β0
and all βj > β0 , j = 1, ..., n we have
−
−
kgr kH→
p (→
α) ≤ c
n
Y
αj
(1 − rj ) pj
−
βj pj −2
pj
,
j=1
for αj > −1 and all 0 < pj < ∞, rj ∈ I, j = 1, ..., n; for pj = ∞ we have
115
βj > 0, j = 1, ..., n and
kgr kH ∞ ≤ C
n
Y
(1 − rj )−βj , rj ∈ (0; 1).
j=1
for all j = 1, ..., n
One of the main goals of this note is to study multipliers of some new H-type
analytic so-called mixed norm spaces with unusual norms and quazinorms. For
that reason first we find direct connections of these new analytic spaces with
analytic function spaces with simpler norms and quazinorms based on various
simple embeddings and then use this connection and already known information
about coefficient multipliers of analytic classes with simpler quazinorms to get
information we search for in standard terms of classical Mp (f, r) function. This
shows again the importance of embeddings connecting various analytic function
spaces in the polydisc. We in particular need these assertions.
→
− −
Lemma 2 (see [1], [2]). Let f ∈ H p (→
α ), 0 < pj ≤ 1, j = 1, .., n, αj > −1,
γj = 2 − pj , j = 1, ..., n. Then we have
Z1
Z1
(1 − rn )αn +γn −1 ...
0
(1 − r2 )α2 +γ2 −1
0
Z1
(1 − r1 )α1 +γ1 −1 ×
0
! pp3
! pp2
1
×M1p1 (r1 , ..., rn , f )r1 dr1
where
Mpp (r1 , ..., rn , f ) =
2
...rn drn ≤ ckf kpHn→
;
−
−
p (→
α)
r2 dr2
Z
|f (r1 ξ1 , ..., rn ξn )|p dξ1 ...dξn ;
Tn
and for all 0 < pj < ∞, αj > −1, j = 1, . . . , n
sup |f (z)|
z∈U n
n
Y
2
(1 − |zj |) pj
+ pα
j
−
−
≤ ckf kH→
p (→
α ).
j=1
From lemma 2 we get
→
− −
Corollary 1. Let f ∈ H p (→
α ), 0 < pj ≤ 1, j = 1, ..., n. Then
M1 (f, r)(1 − r)τ < ∞, r ∈ I,
where τ =
n
P
j=1
(αj − pj + 2) p1j , αj > pj − 2, j = 1, ..., n.
116
Similar result is true for M1 (f, r), r Q
∈ I n . Here we just must replace M1 (f, r)
τ
by M1 (f, r1 , . . . , rn ) and (1 − r) by (1 − rj )τj , where τj pj = αj − pj + 2,
and where rj ∈ I. We will use this corollary and this remark constantly later.
We introduce now new spaces of analytic functions in the polydisc and provide
estimates of the Bergman kernel in quazinorms of these spaces in the polydisc.
Let for all values of indexes 0 < p, q < ∞, γ > 0
Mγp,q = {f ∈ H(U n ) : kf kpMγp,q
=
Z Z1 Z1
0
T
|f (rζ)|q (1 − r)γq−1 dr1 ...drn
pq
dζ < ∞}
0
These spaces were studied recently in [20].
Note simply replacing T by T n we get immediately known analytic Fαp,q
spaces (see [19]). We also define
kf kMγp,∞ =
Z sup |f (rζ)|(1 − r)α
p
! p1
dζ
< ∞, 0 < p < ∞, α ≥ 0.
r∈I n
T
and with usual modification for p = ∞, 0 < q < ∞. We put A/p = 0 if
p = ∞. In the following several lemmas we do not specify parameters p, q
separately meaning they can accept all positive values including ∞. We also do
not put any restriction on parameters γ and s except those which are accepted
in definition of spaces which they represent in the following lemmas.
Lemma 3 (see [20]). Let
gR (z) =
1
, R ∈ I, z ∈ U n .
(1 − Rz)β+1
Then
kgR kMγp,q ≤
kgR kMγp,∞ ≤
C1
1
n(β+1)−γn− p
, β>γ+
1
− 1,
np
C2
1
n(β+1)−γn− p
, β>γ+
1
− 1,
np
(1 − R)
(1 − R)
kgR kMγ∞,q ≤
C3
, β > γ − 1.
(1 − R)(β+1−γ)n
We note Lemma 3 in combination with standard arguments based on socalled closed graph theorem allows to show immediately Theorem 4 below with
the help of lemma 9 and 10. Now we formulate a complete analogue of lemma
3 for Tγp,q spaces in the polydisc.
117
gR (z) =
1
, R ∈ I n , z ∈ U n , β > 0.
(1 − Rz)β+1
Then
kgR kTγp,q ≤
C1
γ
1
β+1− n
−p
(1 − R)
kgR kTγ∞,q ≤
γ
1
+ − 1,
n p
C2
γ
− 1,
γ , β >
β+1−
n
n
(1 − R)
C3
kgR kTγp,∞ ≤
, β>
γ
1
β+1− n
−p
(1 − R)
, β>
γ
1
+ − 1.
n p
The proof of lemma 5 is easy. We omit it leaving it to readers.
n
Q
Lemma 5. For gR (z) =
j=1
−
kgR kH→
p ≤
n
Q
→
−
α
1
,
(1−Rj zj )βj +1
C
βj +1− p1 −αj
(1 − Rj )
R ∈ I n , z ∈ U n , we have
, βj > αj +
j
1
− 1, j = 1, ..., n.
pj
j=1
s
p,q
The following is a complete analogue of lemma 3 for Ap,q
γ , Fγ , H spaces,
namely we provide estimates for the Bergman kernel in the polydisc in these
spaces.
gR (z) =
1
, R ∈ I n , z ∈ U n , β > 0.
(1 − Rz)β+1
Then
kgR kAp,q
≤
γ
kgR kFγp,q ≤
C1
1
β−γ− p
+1
(1 − R)
C2
1
β−γ− p
+1
(1 − R)
kgR kH s ≤
C3
1
(1 − R)β+1− s
, β>γ+
1
− 1,
p
, β>γ+
1
− 1,
p
, β>
1
− 1.
s
The following is a complete analogue of lemma 3 for Ap,∞
, A∞,q
, Fγp,∞ , H ∞ ,
γ
γ
Bl namely we provide estimates for the Bergman kernel in the polydisc in these
spaces.
118
gR (z) =
1
, R ∈ I n , z ∈ U n , β > 0.
(1 − Rz)β+1
Then
kgR kAp,∞
≤
γ
C1
(1 − R)
≤
kgR kA∞,q
γ
kgR kFγp,∞ ≤
1
β−γ− p
+1
, β>γ+
1
− 1,
p
C2
, β > γ − 1,
(1 − R)β−γ+1
C3
(1 − R)
1
β−γ− p
+1
, β>γ+
kgR kBl ≤
C4
, β > −1,
(1 − R)β+1
kgR kH ∞ ≤
C5
, β > −1.
(1 − R)β+1
1
− 1,
p
→
− −
Next lemma plays the same role as lemma 2 for H p (→
α ).
Lemma 8 (see [4]). Let 0 < pj < 1, j = 1, ..., n. Then
Z
|f (z1 , ..., zn |
1
(1 − |zj |) pj
−2
−
dm2n (z) ≤ ckf kH→
p =
j=1
Un
Z
Z
= sup
0<rj ≤1
n
Y
Z
...
T
T
! pp2
! p1
n
1
|f (r1 ζ1 , ..., rn ζn )|p1 d(ζ1 )
...d(ζn )
.
T
We get the following assertion from lemma 8 immediately using standard
arguments.
Corollary 2. Let pj ∈ (0, 1), j = 1, . . . , n, then
sup M1 (f, r)
r∈I n
n
Y
1
(1 − rk ) pk
−1
≤ ckf kH p~
k=1
Corollary 3. Let pj ∈ (0, 1), αj > 0, j = 1, . . . , n, then
sup M1 (f, r)
r∈I n
n
Y
1
(1 − rk ) pk
k=1
−1+αk
≤ ckf kHαp~
119
Weighted Hardy space case Hα~p~ follows from corollary 2 if we apply it to
fR , R ∈ I function using then standard arguments. We add here two important
remarks. As we will see this corollary 2 and it is weighted version we just mentioned can be used immediately instead of corollary 1 to get same theorems for
Hα~p~ instead of H p~ (~
α) for those cases where MT (Y, H p~ (~
α)) appears in discussions
after theorems below, we leave this procedures to readers (we denote here by Y
one of BMOA type spaces we study in this paper). On the other hand Lemma
8 gives ways to find easily simple necessary conditions for g function in usual
→
−
in these issues terms of M1 (f, r) function to be a multiplier from Y to Hα~p for
BMOA type Y spaces we considered above (also with the help of lemmas 3–7
and the closed graph theorem). We will also see this below easily. Our last two
assertions are the most important lemmas for this paper. They connect BMOA
type spaces with much simpler analytic function spaces in the unit polydisc
which leads to solutions of some problems we already indicated for this work.
then
Lemma 9. Let α > 0, s > 0, 0 < q < ∞. Then if f ∈ A∞,q
s
∞,q
f ∈ Fsq−1,α
∞,q
with related estimates for quazinorms and if f ∈ Fsq−1,α
then
f ∈ A∞,∞
s
with related estimates for quazinorms.
Lemma 10. Let f ∈ BM OA then
f ∈ A∞,∞
β,β
for all β > 0. If
∞,∞
f ∈ Aβ,β−1−α
for all β > α + 1 and α > −1 then f ∈ BM OA with related estimates for
quazinorms.
∞,q
The Fs,α
analytic spaces can also be defined in a more general form for
vectors s and α when parameters related with weights are vectors. Most results
∞,q
below concerning this Fs,α
space can be reformulated in this more general form.
We add some comments concerning proofs of these assertions. The first part of
lemma 9 is obvious. For the unit disk case it is known, see, for example, [3]. The
second part of lemma 9 for the case of unit ball can be seen with a complete proof
in [12]. The base of that proof is a Bergman reproducing formula and ForelliRudin type estimates. Note in the polydisc another version of that lemma is
also valid and the proof is a copy of a proof of unit ball case. We omit details
referring the reader to [12] for all details. Turning to the proof of lemma 10 we
120
note the proof of the first part can be seen in [22] and the proof of the second
one follows from the following simple argument. Note first for any α, α > 0 we
1
have f ∈ A1,1
α for any f function so that f ∈ H . Since both classes are Banach
spaces, this means that the reverse inclusion holds for dual spaces. Note now the
dual of the Bergman class is well-known in the polydisc (see, for example, [3])
and hence the assertion of lemma follows. Using last lemmas estimates of the
Bergman kernel and hence gR function can be found for BMOA-type spaces
we introduced in this paper (see also similar assertions above). These simply
are estimates of gR for A∞,∞
or A∞,q
spaces which can already be seen in this
α
α
section. Note now all assertions of next section concerning two scales of BMOA
type functions we study in this paper are in some sence parallel to each other
in a sense that each assertion on BMOA as we defined above leads to similar
q,∞
assertion concerning Fs,α
analytic function spaces with similar arguments for
proofs. We just have to replace embeddings in last two lemmas. This remark can
q,∞
spaces in the polydisc.
be applied to each assertion below formulated for Fs,α
The exposition in this note is sketchy. We alert the reader in advance many
results of this note (and even proofs which are easy calculations) are left to
interested readers, but all details of schemes of proofs for that are indicated
below by us.
3
Main Results on Coefficient Multipliers in some new
spaces of analytic functions of BMOA type in the
polydisc.
The intention of this section is to provide main results of this paper. The
base of all proofs is a standard argument based on closed graph theorem used
before by many authors (see, for example, [3]) in one dimension and in higher
dimension (see, for example, [15], [16], [17], [19] and references there). This in
combination with various lemmas containing estimates of the Bergman kernel
in some new unusual analytic spaces we provided above leads directly to the
completion of proofs. Note again this scheme in U n was used by first author also
in previous mentioned papers [15], [16], [17], [19]. The appropriate definitions of
coefficient multipliers in polydisc and Bergman kernels in the polydisc serve as
important steps for solutions. Our first theorem and the second theorem provide
new necessary conditions for multipliers into various known mixed norm spaces
from BMOA-type spaces which we study in this paper based on first lemmas.
Note these two theorems are not new in case of unit disk (see [3]). So they can
be viewed as far reaching extensions of these old results on coefficient multipliers
of spaces of BMOA-type (see [3]) in the unit disk. The following theorem can
be seen also as a corollary of remark above which provides direct ways to get
estimates of the Bergman kernel in mentioned BMOA-type spaces and estimates
p,q
of the Bergman kernel for Ap,q
spaces. We alert the reader many
α and Fα
121
arguments in this paper are sketchy, this allows using short space to provide big
amount of new results on multipliers of BMOA type spaces in the polydisc. We
also shortly indicate changes for that reason that are needed to provide some
such proofs below to get new assertions.
∞,q
We formulate first two assertions for Fsq−1,α
spaces in this paper.
Theorem 1. 1) Let 0 < p, v < ∞, γ > 0, g ∈ H(U n ). Let also g(z) =
∞
P
∞,q
n
to Ap,v
ck1 ,...,kn z1k1 ...znkn be a multiplier from Fsq−1,α
γ (U ), then
|k|≥0
sup Mp (Dβ g, r)
r∈I n
n
Y
(1 − rj )γ+βj +1−s < ∞,
j=1
for all βj > s − 1 − γ, j = 1, ..., n.
∞
P
2) Let 0 < p, v < ∞, γ > 0, g ∈ H(U n ), g(z) =
k1 ≥0,...,kn ≥0
ck1 ,...,kn z1k1 ...znkn be
∞,q
a multiplier from Fsq−1,α
to Fγp,v (U n ), then for v ≤ p
sup Mp (Dβ g, r)
r∈I n
n
Y
(1 − rj )γ+βj +1−s < ∞,
j=1
for all βj > s − 1 − γ,j = 1, ..., n.
Theorem 2. 1) Let 0 < p, q < ∞, γ > 0, g ∈ H(U n ). Let also g(z) =
∞
P
n
ck1 ,...,kn z1k1 ...znkn , be a multiplier from BM OA to Ap,q
γ (U ) then
|k|≥0
sup Mp (Dβ g, r)
r∈I n
n
Y
(1 − rj )γ+βj +2+v < ∞,
j=1
where β > −v − 2 for any, v, v > −1.
2) Let 0 < p, q < ∞, g ∈ H(U n ), g(z) =
∞
P
k1 ≥0,...,kn ≥0
ck1 ,...,kn z1k1 ...znkn be a
multiplier from BM OA to Fγp,q (U n ), then for q ≤ p
sup Mp (Dβ g, r)
r∈I n
n
Y
(1 − rj )γ+βj +2+v < ∞,
j=1
where β > −v − 2 for any v, v > −1.
The following theorem also can be seen as a corollary of remarks after lemma
9 and 10 above which provides estimates of the Bergman kernel in mentioned
BMOA-type spaces and the closed graph theorem. Let below Y be again the
∞,q
space which we introduced above. The case of BMOA needs small
Fsq−1,α
changes which we already indicated.
122
Theorem 3. Let p, γ > 0. Let g ∈ H(U n ), g(z) =
P
ck z k
|k|≥0
1) If g ∈ MT (Y ; Fγp,∞ ), then
β
sup Mp (D g, r)
r∈I n
n
Y
(1 − rj )γ+βj +1−s < ∞,
j=1
for all βj > s − 1 − γ, j = 1, ..., n.
2) If g ∈ MT (Y ; Ap,∞
), then
γ
sup Mp (Dβ g, r)
r∈I n
n
Y
(1 − rj )γ+βj +1−s < ∞,
j=1
for all βj > s − 1 − γ, j = 1, ..., n.
Remark 1. Similarly conditions on MT (Y, X) for X = Tγp,q and Bγp,q can be
found for p < q or p = q. We have to use that Tγp,q ⊂ Bγp,q , p ≤ q and the fact
that supr∈I Mp (f, r)(1 − r)γ ≤ ckf kBγp,q , γ > 0, 0 < p, q ≤ ∞. These estimates
are easy to see or can be seen in various textbooks on Bergman spaces.
Note that we can also replace Tαp,q spaces in previous remark we just clarified
by Hα~p~ classes. To formulate this new assertion we have to replace one estimate
on Tγp,q spaces we just mentioned by another similar estimate from below via
Mp (f, r) function for Hα~p~ classes. Both can be seen in previous section. For
→
− −
H p (→
α ) spaces the situation is the same and similar arguments can be easily
applied. In both cases we need estimates from below via Mp (f, r) function.
Note changing an embedding we use in proofs of theorems 1, 2 for BMOA by
∞,q
(or reverse lemma 10 by lemma 9) we can easily
similar embedding for Fs,α
∞,q
formulate assertions of theorems 1 and 2 directly in terms of for Fs,α
analytic
classes or by BMOA in the unit polydisc. We leave this last easy procedure to
interested readers.
Proofs of Theorems 1 and 2. Here is a complete scheme of proof of various assertions of type contained in theorem 1 and theorem 2.
Assume {ck }k∈Z+n ∈ MT (Y, X) where
p,q
X = Ap,q
α or Fα , 0 < p, q ≤ ∞, α > 0
or
X = Tαp,q or Mαp,q , 0 < p, q ≤ ∞, α > 0,
or X = Bl, H ∞ .
An application of the closed graph theorem gives kMc f kX ≤ ckf kY .
Let w ∈ U n and set fw (z) =
1
1−wz ,
gw = Mc fw . Then we have
1
D (gw ) = D Mc fw = Mc (D fw ) = c Mc
(1 − wz)β+1
β
β
123
β
; β>0
which gives
1
kDβ gw kX ≤ ckf˜kY = c (1 − wz)β+1 .
Y
This give immediately by lemma above an estimate
kDβ gw kX ≤ c Q
n
1
,
(1 − rj ) − τ0
j=1
for some fixed τ0 depending on parameters involved. We finish easily the proof
using only one known embedding (depending each time on quazinorm of X
space).
For X = Ap,q
α we use embedding
sup (Mp (f, r)(1 − r)α ≤ ckf kAp,q
, f ∈ H(U n ), 0 < p ≤ ∞, α ≥ 0.
α
r∈I n
For X = Fαp,q , p ≤ q we use the same embedding with inclusion
Fαp,q ⊂ Ap,q
α , p ≤ q.
For p = ∞ or q = ∞ we have to replace above p by ∞ or q by ∞. The case if
X = Bl or X = H ∞ we have to modify the scheme we provided above.
For X = Tαp,q we use same arguments and the embedding
sup Mp (f, r)(1 − r)α ≤ ckf kTαp,q ; 0 < p, q ≤ ∞; p ≤ q; α > 0, f ∈ H(U n )
r∈I
with obvious modification for p = ∞ or q = ∞. Note that the same embedding
is true for Bαp,q .
As another corollary of lemmas above we have the following theorems 4,
5 and 6. They provide various necessary conditions for a function to be a
multiplier from well-studied mixed norm analytic Besov type and new LizorkinTriebel type spaces and H-type new Hardy-type and Bergman-type analytic
spaces with mixed norm into new BMOA type spaces in the polydisc we defined
∞,q
and study in this paper namely BMOA and Fα,s
analytic function spaces.
These are far reaching extensions of previously known results (see for unit disk
case [3] and various references there).
124
The crucial embeddings for BMOA type spaces we indicated for these two
spaces at the end of our previous section again as previously in theorems 1, 2
serve as a base of proofs of all our results we will formulate below. Note as in
previous case (theorems 1 and 2) we consider both analytic BMOA type spaces
q,∞
simultaneously, since the case of Fα,s
and BMOA spaces are similar and proofs
are parallel, we only have to change an embedding for BMOA by embedding
q,∞
for Fα,s
which can be seen in lemmas 9 and 10 above.
Note some results contained in theorems below are new even in case of unit
disk, on the other hand others can be viewed as direct extensions of well-known
one dimensional results to the higher dimensional case of the unit polydisc.
Note also here as we had above we include in some cases “limit” spaces when
some parameters involved are equal to infinity. We also leave some cases open
for interested readers since they do not contain any new ideas for proofs. The
following theorem can be seen as a corollary of lemmas above which provides
p,q
estimates of the Bergman kernel in mentioned mixed norm spaces Ap,q
γ , Fγ
and the closed graph theorem and lemma 9 and 10. Various concrete values of
parameter β0 depending on various parameters and on a concrete space can be
also easily recovered by readers in all theorems of this note.
We also leave this to readers.
∞,q1
We formulate below three theorems for Fvq
spaces q1 ∈ (0, ∞), v > 0,
1 −1,α
α > 0. We consider even a little bit more general case of mentioned spaces but
when v is a vector with components vj (see for this a remark above). We will
also denote this space by Y below to avoid big amount of indexes in formulations
of theorems. Note below indexes q1 and α do not appear in spaces of coefficient
multipliers we found. Note also the base of proofs is lemma 9 and the closed
graph theorem. Similar assertions (complete analogues of theorems below) are
valid also for analytic BMOA space we defined above. We have to replace
only lemma 9 by lemma 10 using same arguments in our proofs. We leave this
procedure to interested readers. We assume always below that ab = 0, if b = ∞.
P
Theorem 4. Let p > 0, q > 0, γ > 0 and let g(z) =
ck z k , g ∈ H(U n ).
k∈Zn
+
1) If g ∈
MT (Ap,q
γ ,Y
), then
sup M∞ (Dβ g, r1 , ..., rn )
r∈I n
τj = vj + β − γ −
1
p
n
Y
(1 − rj )τj < ∞,
j=1
+ 1, β > max γ +
j
1
p
− 1 − vj .
2) If g ∈ MT (Fγp,q , Y ), then
sup M∞ (Dβ g, r1 , ..., rn )
r∈I n
n
Y
(1 − rj )τj < ∞,
j=1
τj = vj + β − γ −
1
p
+ 1, j = 1, ..., n, β > max γ +
j
1
p
125
− 1 − vj .
3) If g ∈ MT (Fγp,∞ , Y ), then
sup M∞ (Dβ g, r1 , ..., rn )
r∈I n
τj = vj + β − γ −
1
p
n
Y
(1 − rj )τj < ∞,
j=1
+ 1, j = 1, ..., n, β > max γ +
j
1
p
− 1 − vj .
4)If g ∈ MT (Ap,∞
, Y ) then
γ
sup M∞ (Dβ g, r1 , ..., rn )
r∈I n
τj = vj + β − γ −
1
p
n
Y
(1 − rj )τj < ∞,
j=1
+ 1, j = 1, ..., n, β > max γ +
j
1
p
− 1 − vj .
, Y ) then
5)If g ∈ MT (A∞,q
γ
sup M∞ (Dβ g, r1 , ..., rn )
r∈I n
τj = vj + β − γ −
1
p
n
Y
(1 − rj )τj < ∞,
j=1
+ 1, j = 1, ..., n, β > max γ +
j
1
p
− 1 − vj
with p = ∞.
The following two theorems can be seen as a corollary of lemmas 3 and 4, and
lemmas 9 and 10 above. Lemma 3 and 4 provides estimates of the Bergman
kernel in mentioned mixed norm spaces Tγp,q , Mγp,q including limit “infinity”
cases of parameters and the closed graph theorem. Note similar results are valid
for Bαp,q spaces. We leave this to readers. To get estimates for the Bergman
kernel for B p,q spaces we refer the reader to [20]. We assume again Y is a same
BMOA space as in previous theorem.
P
Theorem 5. Let p, q, γ > 0, v > 0. Let g ∈ H(U n ), g(z) =
ck z k .
n
k∈Z+
1
1) If g ∈ (MT )(Mγp,q , Y ), then sup M∞ (Dβ g, r)(1 − r)v+n(β+1)−γn− p < ∞, for
r∈I
β>γ+
1
np
−
v
n
− 1.
1
2) If g ∈ (MT )(Mγp,∞ , Y ), then sup M∞ (Dβ g, r)(1 − r)v+n(β+1)−γn− p < ∞, for
r∈I
β>γ+
1
np
−
v
n
− 1.
3) If g ∈ (MT )(Mγ∞,q , Y ), then sup M∞ (Dβ g, r)(1 − r)v+n(β+1)−γn < ∞, for
β>γ−
v
n
r∈I
− 1.
126
Replacing Mγp,q , Mγp,∞ and Mγ∞,q by Tγp,q , Tγp,∞ and Tγ∞,q , respectively, we
have the following analogue of previous theorem for Tαp,q type spaces of analytic
function in the polydisc based again on lemmas 3, 4 and 9, 10 we provided
above. Note these results though are not sharp but are new even in case of unit
disk. We formulate this theorem the last one for Fτ∞,d
d−1,α classes, for this case
also d parameter and α parameter do not appear in spaces of multipliers we
found below. We note that we consider general case when τ is a vector (see for
this general case our discussion above).
P
Theorem 6. Let p, q, γ > 0. Let g ∈ H(U n ), g(z) =
ck z k .
n
k∈Z+
n
Q
γ
1
(1 − rj )τj +β+1− n − p < ∞,
1) If g ∈ (MT )(Tγp,q , Y ), then sup M∞ (Dβ g, r)
n
r∈I
j=1
β > max p1 + nγ − τj − 1 , τj > 0; j = 1, . . . , n.
j
n
γ
Q
1
(1 − rj )τj +β+1− n − p < ∞,
2) If g ∈ (MT )(Tγp,∞ , Y ), then sup M∞ (Dβ g, r)
n
r∈I
j=1
γ
1
β > max p + n − τj − 1 , τj > 0; j = 1, . . . , n.
j
n
γ
Q
1
(1 − rj )τj +β+1− n − p < ∞,
3) If g ∈ (MT )(Tγ∞,q , Y ), then sup M∞ (Dβ g, r)
n
r∈I
j=1
γ
1
β > max p + n − τj − 1 , τj > 0, j = 1, . . . , n with p = ∞.
j
We add another short comment here. Note that we can also replace Tαp,q
spaces in previous theorem by Hα~p~ classes. To formulate such type new assertion
we have to replace estimates of the Bergman kernel of Tαp,q spaces by estimates
of Bergman kernel (gR function) in quazinorms of Hα~p~ classes. Both can be seen
→
− −
in previous section. For H p (→
α ) spaces similar arguments can be easily applied.
→
− −
All results needed for proofs for H p (→
α ) and Hα~p~ analytic classes can be found
in the previous section. We leave this procedure to for interested readers.
Here is a short scheme of proof of various assertions of type contained in
theorems 4, 5 and 6. Assume {ck }k∈Z+n ∈ MT (X, Y ) then we have to use the
closed graph theorem and lemma 9, 10. Indeed we have the following estimates
and equalities. First by closed graph theorem kMc f kY ≤ ckf kX . Let w ∈ U n ,
1
and set fw (z) = 1−wz
, gw = Mc (fw ). Then we have
Dβ gw = Dβ (Mc (fw )) = Mc (Dβ (fw )) = cMc
1
, β > 0,
(1 − wz)β+1
which gives
1
.
kD gw kY ≤ c β+1
(1 − wz)
β
X
127
It remains to apply lemma 9, 10 from left for BMOA type spaces from right for
various X spaces we have to use various estimates from lemma 3, 4, 6, 7 of the
Bergman kernel gR (z) in the polydisc U n to get the desired result.
We finally add one more remark.
Remark 2. Note probably some assertions of theorems 4 – 6 can be sharpened,
namely the reverse implications can be proved also. We do not discuss these
issues in this paper.
We can also use for some new assertions the following simple idea (see [3],
[10], [11]), the following obvious fact if (g) ∼ (ck ) is a multiplier from X to Y
(X, Y are quazynormed subspaces of H(U n )) then it is a multiplier for dual
spaces from Y ∗ to X ∗ and then apply to this last pair a standard closed graph
theorem. This procedure was used before by many authors (see, for example,
[3]). We can also use this type argument for our purposes. Descriptions of duals
→
− −
of H p (→
α ) spaces for pj > 1, j = 1, ..., n or pj < 1, j = 1, ..., n are known (see
p,q ∗
) , (Apαe,eq )∗ spaces are also
for this description [1], [2]). Note also (H p )∗ , (Aα
well-known (see, for example, [3]).
References
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functions, Investigations on linear operators and function theory. 26, Zap.
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R. Shamoyan
Department of Mathematics, Bryansk State University
Bryansk, 241050, Russia
[email protected]
S. Kurilenko
Department of Mathematics, Bryansk State University
Novozibkov, 241098, Russia
[email protected]
DIVULGACIÓN CIENTÍFICA
Una extraordinaria acotación inferior
Luis Gómez Sánchez*
La desigualdad π − pq > q142 , válida para todo racional pq con q ≥ 2 no
es reciente, se debe a Mahler (1903-1988) y fue publicada en 1953 por partida
doble en The Philosophical Transactions of the Royal Society A, 245 (Inglaterra), y en Indagationes Mathematicae, 15 de la Royal Dutch Mathematical
Society (Holanda). Esta desigualdad, de un carácter inusual hasta la fecha de
su descubrimiento, indica (con las limitaciones de su tiempo, como veremos al
final) de qué modo los racionales no pueden acercarse a π: todo racional
“cercano” a π determina su “alejamiento” de π, en el sentido de nunca estar
a una distancia menor o igual que la potencia −42 de su denominador. Esto,
aparentemente, irı́a en contra del hecho básico y muy conocido del análisis real
Nn
de que todo irracional α es un lı́mite de racionales D
(Q es denso en R), pero
n
es fácil ver que no es ası́.
¿De dónde proviene esta desigualdad notable? (“striking inequality” comenta Alan Baker, Medalla Fields y gran teórico de los números trascendentes).
Situaremos en un contexto algo más informativo, aunque de manera bastante abreviada, esta atrayente desigualdad de Mahler, quien por cierto también
hizo otros aportes importantes en teorı́a de números, en particular una sutil
clasificación en cuatro clases, algebraicamente independientes, de los números
complejos muy útil en relación con los números trascendentes y que ha sido
fuente de problemas abiertos de muy grande dificultad. También demostró que
el número 0,1234567891011 · · · , cuya parte decimal sigue la sucesión de enteros
naturales, es trascendente.
Nn
= α, entonces tanto el numeSuele pasar desapercibido que si lı́mn→+∞ D
n
rador Nn como el denominador
D
deben
crecer
desmesuradamente (salvo si
n
Nn α es racional); entonces, si α − Dn < no puede haber ningún control de la
relación entre las tallas de y de Dn . Se plantea entonces el problema de establecer qué tan buena puede ser una aproximación del irracional α por medio
de un racional pq , digamos para precisar mejor un racional pq “no inaccesible”,
* Dedico con todo cariño este artı́culo a la XLIX Semana Aniversaria del Departamento
de Matemáticas (11 al 15 de noviembre 2013). Escuela de Ciencias. Universidad de Oriente.
Núcleo de Sucre. Cumaná, Venezuela.
132
Luis Gómez Sánchez
Nn ya que es obvio que α − D
puede ser tan pequeño como se quiera pero con
n
racionales que no están al alcance de la mano. Este problema está a la base
de la llamada Aproximación Diofántica (por el mismo griego, Diofanto, de las
conocidas ecuaciones diofánticas, es decir sistemas, como por ejemplo x + y = 1,
con más variables que ecuaciones) la cual está en estrecha relación con la difı́cil
teorı́a de números trascendentes cultivada minoritariamente por los matemáticos. Pero esta clase de aproximación difiere fundamentalmente de aquella del
simple cálculo aproximativo, más que la solución aproximada de una ecuación
le interesa resolver desigualdades en números enteros (también ecuaciones, en
particular la finitud del número posible de soluciones). Veamos algo mejor el
punto usando al mismo número π.
22
La doble acotación 223
71 < π < 7 , obtenida por Arquı́medes (siglo III a.C.),
da para π la aproximación escolar muy usada de 22
7 con dos cifras decimales
37
88
correctas. Y un racional de no tan antigua procedencia lo da la suma 22
17 + 47 + 83
con 9 cifras decimales correctas de π. Pero estas aproximaciones, que permiten
en ciertos contextos reemplazar π por las mismas tienen una filosofı́a distinta
de
anima
q la que
qlas aproximaciones diofánticas y ası́ por ejemplo los números
√
40
12 y 9 34041350274878
3 −
1141978491 , que dan respectivamente 4 y 15 cifras exactas
para π por lo cual tienen un mayor interés “aproximativo” que 22
7 , no interesan
aquı́ por no ser racionales; de hecho, no interesa ningún racional en particular,
37
88
por más que mejore la “buena” aproximación dada por 22
17 + 47 + 83 , y no se
busca de reemplazar π por ningún valor aproximado. Lo que interesa más bien
es la llamada medida de irracionalidad. Se demuestra que la de los racionales
es igual a 1, la de los irracionales algebraicos es igual a 2 (¡gran descubrimiento
de Klaus Roth!) y la de los trascendentes es mayor o igual que 2, en particular
la de los números de Liouville (que mencionaremos al paso más adelante) es
infinita y la de la base e de los logaritmos naturales es exactamente 2, tal como
si e fuera algebraico y no trascendente.
Definición. La medida de irracionalidad de un real α es el ı́nfimo de todos
los reales µ para los cuales existe una constante
positiva A tal que para todo
racional pq 6= α con q 6= 0 se tiene α − pq > qAµ .
Esta definición, que varı́a según los autores aunque de forma en general
equivalente, equivale a que, σ denotando dicho ı́nfimo, para todo > 0 y todo
racional pq 6= α con denominador suficientemente grande se verifica la desigual
1
1
dad α − pq > qσ+
, o aún que para todo > 0 la desigualdad α − pq < qσ+
tiene sólo un número finito de soluciones en racionales irreductibles
final comentario sobre el teorema de Roth).
p
q
(ver al
Ejemplo. Baker, hacia 1980, después de exponer un teorema general escribe
133
lo siguiente: “en particular, eπ − pq > q −c log log q para todo racional pq (q > 4)
donde c denota una constante absoluta, y esta es la mejor medida de irracionalidad para eπ obtenida hasta la fecha”.
Nota. Salvo error de información, esta medida no ha sido aún mejorada ni se
sabe si log log q es el ı́nfimo con el cual se cerrarı́a el muy difı́cil problema de la
medida de irracionalidad de la constante de Gelfond eπ que, como se sabe, es
un número trascendente.
Definición. Sea α un número real; el racional pq es llamado una buena apro
0
0
ximación diofántica de α si α − pq ≤ α − pq0 para todo racional pq0 tal que
0 < q 0 ≤ q, es decir, si la diferencia entre α y pq no decrece cuando este racional
se cambia por cualquier
positivo menor que q.
p0
q0
con numerador arbitrario y cualquier denominador
Nota. El principiante se puede preguntar por qué esta consideración exclusiva
del denominador solamente, con exclusión del numerador; la respuesta se la
podrı́a dar él mismo al razonar un poco sobre el punto.
Conociendo una buena aproximación del irracional α (idealmente “la mejor”),
se presenta de hallar refinadas cotas superiores e inferiores
el problema
p
de α − q expresadas como función del denominador y constantes, sean éstas
efectivas (es decir o calculables o susceptibles de explicitarse como algoritmo)
o no efectivas, en la cuales no se establece su valor como constantes absolutas
y sólo se demuestra su existencia. La historia de esto es amplia y la resumimos
aquı́ de manera abreviada, como ya dijimos.
El lunes 13 de mayo de 1844, Joseph Liouville (1809-1882) presentó a la Academia de Ciencias de Parı́s dos informes, uno de matemáticas y otro de fı́sica.
El primero de estos Comptes-rendus, constituye un pináculo en la historia de las
matemáticas: Liouville estableció “des classes très étendues de quantités dont
la valeur n’est ni rationnelle ni même réductible à des irrationnelles algébriques”, “une grande quantité de nombres qui ne sont pas algébriques”. Ese dı́a
∞
X
Liouville expuso el primer número trascendente conocido, el
10−k! en cuyo
k=1
desarrollo decimal el 1, única cifra no nula que interviene, se va enrareciendo
muy rápidamente con el crecimiento rápido del factorial (cada número de la
“grande quantité” que descubrió recibe el nombre de número de Liouville). El
hecho era un gran suceso porque el conjunto de todos los números algebraicos
(reales) es numerable y en consecuencia su complemento en R, el conjunto de
los trascendentes (reales), tiene la potencia del continuo pero hasta esa fecha no
se conocı́a ningún trascendente, a pesar de que habı́an “muchos más” trascendentes que algebraicos (como se sabe, los algebraicos son los números que son
134
raı́ces de polinomios a coeficientes enteros). Pues bien, Liouville hizo un trabajo
de aproximación diofántica para lograr su gran descubrimiento, en particular
usó una propiedad elemental de las fracciones continuas las
ofrecen de
cuales
p
modo simple una doble acotación, inferior y superior, para α − q .
Teorema (Liouville). Para todo número
α, real de grado n ≥ 2
algebraico
c(α)
p
existe un número positivo c(α) tal que α − q > qn se cumple para todos los
racionales pq , q > 0.
Es decir, la medida de irracionalidad del irracional algebraico α no puede ser
mayor que su grado (es bastante menor pero habı́a que esperar hasta 1955 para
saberlo). Fue este teorema el que permitió definir explı́citamente los números
trascendentes de Liouville.
Definición. Se dice que el número real irracional α es un número de Liouville
si para todo n suficientemente grande existe un racional pq con q > 1 tal que
α − pq < q1n .
Nótese que, por la definición misma, todo número de Liouville tiene una
aproximación “muy fina” por racionales.
Todo número de Liouville es trascendente. Por definición se ve que la
medida de irracionalidad de todo número de Liouville es infinita.
Damos para concluir una sucinta reseña de los progresos obtenidos en la
dirección de este teorema, es decir, sobre la medida de irracionalidad de los
reales algebraicos. Veremos que se logró una notable reducción, desde el grado
n (que puede ser tan grande como se desee) hasta 2, independientemente del
grado. Damos los teoremas bajo su forma de aproximación diofántica del número
algebraico α por un número finito de racionales “accesibles” (nos referimos a
estos como “no inaccesibles” al comienzo).
• Joseph Liouville (1809-1882), francés: aproximación n + , en 1844.
• Axel Thue (1863-1922), noruego: aproximación
n
2
+ 1 + , en 1909.
√
• Carl Ludwig Siegel (1896-1981), alemán: aproximación 2 n + , en 1929.
Siegel conjeturó la mejor cota hallada después por Roth.
√
• Freeman John Dyson (1923- ), británico-americano: aproximación 2n + ,
en 1947.
• Klaus Friedrich Roth (1925- ), germano-británico: aproximación 2+, en 1955.
La mejor cota, un resultado profundo que le valió a Roth la Medalla Fields.
“The achievement is one that speacks for itself: it closes a chapter, and a new
135
chapter is now opened. Roth’s theorem settles a question which is both of a
fundamental nature and of extreme difficulty. It will stand as a landmark in
mathematics for as long as mathematics is cultivated” (Harold Davenport, de
su presentación de Roth para la Medalla Fields en el Congreso Internacional
de Edimburgo, 1958).
El teorema de Roth
que para todo irracional algebraico α y todo
asegura
p
1
tiene sólo un número finito de soluciones
> 0 la desigualdad α − q < q2+
p
en racionales irreductibles q , lo que da como consecuencia que para todo > 0,
existe una constante positiva C(α, ) tal que la desigualdad α − pq > C(α,)
q 2+ se
p
cumple para todos los racionales q con q > 0.
Nótese que la constante C(α, ) no es efectiva, que sólo se demuestra su
existencia; nótese también que, como dicho al comienzo con la desigualdad de
Mahler para el trascendente π, los racionales no pueden aproximarse “controladamente” a un irracional algebraico α: con Liouville quedaban “no muy cercanos
de α” pero con Roth quedan “bastante más lejos de α”.
Ejercicio. Trasladar adecuadamente lo dicho sobre el teorema de Roth, a los
sendos teoremas indicados de Liouville, Thue, Siegel y Dyson.
Nota. Es pertinente subrayar que (con miras a una aproximación diofántica por
un número finito de racionales) el exponente 2+ en el teorema de Roth debe ser
con > 0, no importa lo pequeño que sea; ası́ 2 satisface la definición de medida
de irracionalidad para el irracional algebraico y arbitrario α. Sin embargo el
teorema deja de ser válido si = 0 en cuyo caso por el antiguo teorema de
aproximación (también diofántica) de Dirichlet
(1842), hay una infinidad de
racionales pq que verifican la desigualdad α − pq < q12 para todo irracional α,
aunque no sea algebraico (todos los convergentes pq del desarrollo en fracción
continua de α satisfacen esta desigualdad). Algo similar sucede con la serie
∞
X
1
, que converge para todo > 0; pero si = 0 se tiene la conocida serie
1+
n
n=1
armónica de las inversas de los enteros, la cual es sabido que diverge; por lo
demás ejemplos análogos no escasean en matemáticas.
¿La desigualdad vista aquı́ de Mahler afirma acaso que la medida de irracionalidad de π es igual a 42? La respuesta es no. Y tampoco se ha demostrado
que la notable mejorı́a con el valor 7.6063, obtenida por el matemático ruso
Salikhov en el 2008, lo sea. ¿Será igual a 2 como para e y para todo irracional
algebraico? Por lo pronto Alekseyev ha demostrado en el 2011 que si la serie
∞
X
csc2 n
converge, entonces eso implicarı́a que la medida de irracionalidad de
n3
n=1
136
π serı́a menor o igual que 2.5. Empero, el cuadrado de la elemental cosecante
de n (que en muchas ocasiones tendrá valores enormes) dificulta las cosas y el
problema de la convergencia de esta serie permanece abierto.
Consultar “http://mathworld.wolfram.com/IrrationalityMeasure.htm”
para una breve mención de estos dos importantes descubrimientos y referencias.
Sobre el tópico de la aproximación diofántica y sus implicancias se puede
decir mucho más que en una simple nota divulgativa y a veces de modo muy
sugestivo aunque no fácil de establecer; por ejemplo el gran Erdös demostró que
todo número real puede escribirse como suma y como producto de dos números
de Liouville lo que reviene a decir que todo real es una suma y un producto
de dos trascendentes especiales ya que un trascendente no es necesariamente de
Liouville. De otro modo la cosa serı́a algo trivial (¿por qué?).
Referencias
[1] Alan Baker. Trascendental Number Theory, Cambridge University Press
1979.
[2] Alan Baker, Breve introducción a la teorı́a de números, Alianza Editorial
S. A. Madrid 1986
[3] Alekseyev, M. A. On Convergence
http://arxiv.org/abs/1104.5100/. 2011.
of
the
Flint
Hills
Series.
[4] Paul Erdös, Representations of real numbers as sums and products of Liouville numbers, Michigan Math. J., Vol. 9, No. 1, 1962
Departamento de Matemáticas, Escuela de Ciencias
Universidad de Oriente. Núcleo Sucre
Cumaná, Venezuela
Boletín de la Asociación Matemática Venezolana, Vol. XXI, No. 2 (2014) 137
INFORMACIÓN NACIONAL
La Esquina Olímpica
Rafael Sánchez Lamoneda
Esta Esquina reseña toda la actividad del año 2014. De Enero a Junio se
llevó a cabo la décimo primera versión de la Olimpiada Juvenil de Matemáticas
(OJM). En la misma participaron 56.898 jóvenes provenientes de 22 ciudades,
representando a 433 colegios y liceos del país. Estos números se desglosan de la
siguiente manera, 183 colegios privados y 248 liceos públicos, con 17.835 alumnos de los colegios privados y 39.063 de los liceos. Como en años anteriores
la primera fase de la OJM fue el Canguro Matemático, y este año en todo el
mundo participaron 6.084.530 jóvenes y 56 países, lo que convierte al Canguro en la competencia de matemáticas escolares, internacional y presencial, más
grande del mundo. Sumando las cifras de participación de la OJM y la Olimpiada Recreativa de Matemáticas, en nuestro país tomaron la prueba Canguro
146.793 álumnos de tercer grado de primaria a quinto año de bachillerato, lo
que nos permitió ocupar el puesto número cinco, en cantidad de participantes.
Tan importante logro se debe al apoyo que recibimos de la Fundación Empresas
Polar.
En la escena internacional tuvimos un año extraordinario, nuestros muchachos lograron ganar 25 premios, entre medallas y menciones honoríficas. Como
en años anteriores asistimos a tres eventos internacionales presenciales y participamos por correspondencia en uno. Los internacionales presenciales fueron,
la 55a Olimpiada Internacional de Matemáticas (IMO), celebrada en Ciudad
del Cabo, Sur Africa, del 3 al 13 de Julio. Participamos con dos estudiantes,
Rafael Aznar del colegio Los Arcos de Caracas y José Tomás Guevara, del colegio Bella Vista de Maracay. Ambos ganaron mención honorífica. La tutora
de la delegación fue Sofía Taylor, estudiante de la licenciatura en Física de la
Universidad Central de Venezuela (UCV), y el jefe de delegación el Prof. Rafael
Sánchez, de la Escuela de Matemáticas de la UCV. Rafael Aznar y José Tomás
Guevara también nos representaron en la XXIX Olimpiada Iberoamericana de
Matemáticas, (OIM), celebrada en San Pedro Sula, Honduras, del 19 al 27 de
Septiembre. Ambos ganaron medalla de bronce por su destacada actuación. La
delegación fue liderada por Estefanía Ordaz, alumna de la licenciatura en Matemáticas de la Universidad Simón Bolívar, y la tutora fue Sofía Taylor. El tercer
evento presencial fue la XVI Olimpiada Matemática de Centroamérica y El Caribe, celebrada en San José de Costa Rica del 6 al 14 de Junio. Nuestro equipo
138
estuvo conformado por tres jóvenes: Amanda Vanegas, del colegio San Francisco de Asís, Maracaibo, Iván Rodríguez del Santiago de León, Caracas y Wemp
Pacheco, del colegio Calicantina de Maracay. Los tres ganaron menciones honoríficas. El jefe de la delegación fue el prof José H Nieto de La Universidad del
Zulia, y la tutora Estefanía Ordaz. Cabe destacar que el profesor Nieto asitió a
la XXIX OIM, como miembro de los tribunales de calificación y fue el encargado
de producir el banco de problemas de la competencia, todo esto por invitación
de los organizadores. El evento por correspondecia fue la XX Olimpiada Matemática de Mayo, que como hemos explicado otras veces, es una competencia
iberoamericana, organizada por Argentina. Participamos con 8 alumnos en el
primer nivel y 10 en el segundo, el máximo es diez en cada nivel y obtuvimos una
medalla de oro, cuatro de plata, ocho de bronce y cinco menciones honoríficas.
Todos nuestros estudiantes, en las cuatro competencias, ganaron algún premio.
No quiero dejar pasar el gran esfuerzo que realizan nuestros entrenadores, principalmente estudiantes universitarios, ex olímpicos, muy bien dirigidos por Sofía
Taylor y Estefanía Ordaz. Toda la información dada se puede complementar en
nuestro sitio de internet, www.acm.ciens.ucv.ve, allí encontrarán los exámenes
de la OJM, así como material de estudio para entrenamiento de estudiantes.
Finalizamos con las dos pruebas de la 55a IMO. Como siempre cada problema tiene un valor de 7 puntos y el tiempo para resolver cada examen es de 4
horas y media.
OLIMPIADA INTERNACIONAL DE MATEMÁTICAS
CIUDAD DEL CABO, SUR AFRICA, 8 de julio de 2014
Primer Día
Problema 1. Sea a0 < a1 < a2 < · · · una sucesión infinita de números enteros
positivos. Demostrar que existe un único entero n ≥ 1 tal que
an <
a0 + a1 + · · · + an
≤ an+1 .
n
Problema 2. Sea n ≥ 2 un entero. Consideremos un tablero de tamaño n ×
n formado por n2 cuadrados unitarios. Una configuración de n fichas en este
tablero se dice que es pacífica si en cada fila y en cada columna hay exactamente
una ficha. Hallar el mayor entero positivo k tal que, para cada configuración
pacífica de n fichas, existe un cuadrado de tamaño k × k sin fichas en sus k 2
cuadrados unitarios.
Problema 3. En el cuadrilátero convexo ABCD , se tiene ∠ABC = ∠CDA =
90◦ . La perpendicular a BD desde A corta a BD en el punto H. Los puntos S y
La Esquina Olímpica
139
T están en los lados AB y AD, respectivamente, y son tales que H está dentro
del triángulo SCT y
∠CHS − ∠CSB = 90◦ ,
∠T HC − ∠DT C = 90◦ .
Demostrar que la recta BD es tangente a la circunferencia circunscrita del
triángulo T SH
OLIMPIADA INTERNACIONAL DE MATEMÁTICAS
CIUDAD DEL CABO, SUR AFRICA, 9 de julio de 2014
Segundo Día
Problema 4. Los puntos P y Q están en el lado BC del triángulo acutángulo
ABC de modo que ∠P AB = ∠BCA y ∠CAQ = ∠ABC. Los puntos M y N
están en las rectas AP y AQ, respectivamente, de modo que P es el punto medio
de AM , y Q es el punto medio de AN . Demostrar que las rectas BM y CN se
cortan en la circunferencia circunscrita del triángulo ABC.
Problema 5. Para cada entero positivo n, el Banco de Ciudad del Cabo
produce monedas de valor n1 . Dada una colección finita de tales monedas (no
necesariamente de distintos valores) cuyo valor total no supera 99+ 12 , demostrar
que es posible separar esta colección en 100 o menos montones, de modo que el
valor total de cada montón sea como máximo 1.
Problema 6. Un conjunto de rectas en el plano está en posición general si no
hay dos que sean paralelas ni tres que pasen por el mismo punto. Un conjunto
de rectas en posición general separa el plano en regiones, algunas de las cuales
tienen área finita; a estas las llamamos sus regiones finitas. Demostrar que para
cada n suficientemente grande, en cualquier conjunto
de n rectas en posición
√
general es posible colorear de azul al menos n de ellas de tal manera que
ninguna de sus regiones finitas tenga todos los lados de su frontera azules.
√
√
Nota: A las soluciones que reemplacen n por c n se les otorgarán puntos
dependiendo del valor de c.
Escuela de Matemáticas, Facultad de Ciencias
Universidad Central de Venezuela, Caracas.
[email protected]
Universidad Central de Venezuela
XXVIII Jornadas Venezolanas de Matemáticas
Caracas, 23 al 26 de marzo de 2015
Llamado a presentación de trabajos
Las Jornadas Venezolanas de Matemáticas son organizadas anualmente por la
Asociación Matemática Venezolana en colaboración con universidades nacionales. Se trata de un evento de carácter cientı́fico que sirve como mecanismo de
registro y divulgación de los resultados de las investigaciones en Matemática
que se realizan en el paı́s. Constituyen un foro de encuentro de matemáticos y
diversos profesionales interesados en conocer, analizar y debatir los temas más
actuales en investigación matemática. La vigésima octava edición de las Jornadas Venezolanas de Matemáticas se efectuará en la Facultad de Ciencias de la
Universidad Central de Venezuela, en Caracas, del 23 al 26 de marzo de 2015.
Las sesiones temáticas del evento y sus coordinadores, son:
Sesión
Coordinadores
Álgebra
y Teorı́a de Números
José Gregorio Fernandes (USB), [email protected]
Aurora Olivieri (USB), [email protected]
Análisis
Análisis Numérico
y Computo Cientı́fico
Ángel Padilla (UCV), [email protected]
Maricarmen Andrade (UCV),
Ecuaciones Diferenciales
y Análisis de Clifford
Luis Mármol (USB), [email protected]
Eusebio Ariza (USB), [email protected]
Javier Villamizar (USB), [email protected]
Milagros Rodrı́guez (UDO), [email protected]
Lysis González, [email protected]
Maira Valera (UCV), [email protected]
David Coronado (USB), [email protected]
Domingo Quiroz (USB), [email protected]
Daniela Torrealba (UCV), [email protected]
Nerio Borges (USB), [email protected]
Ricardo Rı́os (UCV), [email protected]
José Hernández (UCV), [email protected]
Hanzel Lárez (ULA), [email protected]
Carlos Carpintero (UDO), [email protected]
Ennis Rosas (UDO), [email protected]
Educación Matemática
Funciones de variación
acotada y convexas
Grafos y Combinatoria
Lógica Matemática
Probabilidad
y Estadı́stica
Sistemas Dinámicos
Topologı́a y Geometrı́a
[email protected]
Juan Guevara (UCV), [email protected]
142
XXVIII Jornadas Venezolanas de Matemáticas
Las personas interesadas en participar como expositores en cualquiera de las
sesiones temáticas deberán atender a los siguientes requerimientos:
(1) La propuesta de toda comunicación oral en cualquier sesión temática tiene
que ser hecha ante la coordinación de la correspondiente sesión. El resumen
de la misma, de no más de una cuartilla, deber ser enviado, en archivos .tex
y .pdf, a las direcciones electrónicas de los coordinadores de la sesión. La
fecha lı́mite para ello es 30 de enero de 2015.
(2) La selección de las comunicaciones orales de cada sesión es competencia de
la coordinación de la misma. Los proponentes recibirán, antes del 10 de
febrero de 2015, la notificación sobre la aceptación o no de su propuesta.
(3) Toda comunicación oral en las sesiones tendrá una duración de 20 minutos. El número máximo de ellas es de 30; sus horarios serán establecidos
por el Comité Organizador, mientras que el orden de presentación será competencia de la coordinación de cada sesión.
(4) Los resúmenes deberán ser sometidos siguiendo el siguiente formato LATEX:
\documentclass[12pt]{amsart}
\usepackage[spanish]{babel}
\begin{document}
\title{TITULO DE LA COMUNICACION ORAL}
\author{Autor 1, \underline{Autor 2}} %% subrayado el expositor
\address{INSTITUCION del Autor 1}
\email{[email protected]} %% direcci\’on electr\’onica del Autor 1
\address{INSTITUCION del Autor 2}
\email{[email protected]} %% direcci\’on electr\’onica del Autor 2
\maketitle
\section*{Resumen}
ACA VIENE EL CONTENIDO DEL RESUMEN
\begin{thebibliography}{99}
\bibitem{zw}
Z. Zhou and J. Wu. Attractive Periodic Orbits in Nonlinear
Discrete-time Neural Networks with Delayed Feedback.
\textit{J. Difference. Equ. and Appl.} Vol. {\bf 8},
(2001) 467--483.
\end{thebibliography}
\end{document}
AGRADECIMIENTO
Agradecemos la colaboración prestada por las siguientes personas en el trabajo Editorial del Volumen XXI del Boletı́n de la AMV: José Giménez, Dolores
Morán y Rafael Oswaldo Ruggiero.
El Boletı́n de la Asociación Matemática Venezolana está dirigido a un público
matemático general que incluye investigadores, profesores y estudiantes de todos los
niveles de la enseñanza, además de profesionales de la matemática en cualquier espacio del mundo laboral. Son bienvenidos artı́culos originales de investigación en
cualquier área de la matemática; artı́culos de revisión sobre alguna especialidad de la
matemática, su historia o filosofı́a, o sobre educación matemática. El idioma oficial es
el español, pero también se aceptan contribuciones en inglés, francés o portugués.
Todas las contribuciones serán cuidadosamente arbitradas.
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Volumen XXI, Número 2, Año 2014
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EDITORIAL
La Medalla Fields 2014
Oswaldo Araujo
67
ARTÍCULOS
On Almost Pseudo Quasi-Conformally Symmetric Manifolds.
69
Estabilidad y regularidad de operadores.
87
Coefficient bounds for subclasses of bi-univalent functions defined by
the Sălăgean derivative operator.
97
On multipliers of some new analytic function spaces of BMOA type
in the polydisc.
109
DIVULGACIÓN CIENTÍFICA
Una extraordinaria acotación inferior.
131
INFORMACIÓN NACIONAL
La Esquina Olı́mpica.
Rafael Sánchez Lamoneda
137
XXVIII Jornadas Venezolanas de Matemáticas.
141
Agradecimiento.
143

BAMV-XXI-2 - ivic.gob.ve

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