Why we solve the operator equation AX − XB = C ∗†‡

Why we solve the operator equation
AX − XB = C ∗†‡
Salah Mecheri1
Abstract
This work studies how certain problems in quantum theory have
motivated some recent reseach in pure Mathematics in matrix and
operator theory. The mathematical key is that of a commutator
or a generalized commutator, that is, find an operator X ∈ B(H)
satisfying the operator equation AX − XB = C. By this we will
show how and why to solve the operator equation AX − XB = C.
Some problems are studied and some open questions are also given.
1
Introduction
Let B(H) be the algebra of all bounded linear operators on a separable
infinite dimensional complex Hilbert space H. This work studies how certain
problems in quantum theory have motivated some recent reseach in pure
Mathematics in matrix and operator theory. The mathematical key is that
of a commutator. Given A, B ∈ B(H). The operator C is said to be a
commutator, if there exists an operator X ∈ B(H) such that AX −XA = C.
In general, if there exists an operator X ∈ B(H) such that AX − XB =
C, then C is said to be a generalized commutator. The first important
contribution to the study of commutators is due to A. Wintner who in 1947
∗
Key words: Operators equation, Commutator, Putnam-Fuglede’s theorem.
2000 Mathematics Subject Classification: Primary 47A30, 47B47; Secondary
47A15, 47A63
‡
This work was supported by the research center project No. Math/2007/10
1
College of Science, Department of Mathematics, P.O. Box 2455, Riyadh 11451, Saudi
Arabia. e-mail: [email protected]
†
1
2
Why to solve the operator equation AX − XB = C
proved that the identity operator I is not a commutator, that is, there are
no element X such that
I = AX − XA
(1.1)
(to see this, just take the trace of both sides of (1.1)). Nor can (1.1) hold
for bounded linear operators A and X : two nice proofs of this are due to
Wielandt and A. Wintner[17]. Like much good mathematicians, Wintner’s
theorem has its roots in physics. Indeed, it was prompted by the fact that
the unbounded linear maps P and Q representing the quantum-mechanical
momentum and position, respectively, satisfy the commutation relation
P Q − QP = (
−ih
)I,
2π
where h is the Planck’s constant and I is the identity operator. Actually one of the preocupations is the structure of a commutator and a noncommutator. For this it is very intersting to solve the operator equation
AX − XB = C. In [14] W.E. Roth has shown for finite matrices A and B
over a field that AX − XB = C is solvable for X if and only if the matrices
"
A 0
0 B
#
"
and
A B
0 C
#
are similar. A considerably briefer proof has been given by Flanders and
Wimmer [4]. In [13] Rosenblum showed that the result remains true when A
and B are bounded selfadjoint operators in B(H). In this note we will generalize these results for the case where A is normal and (A, B) (resp(B, A))
satisfies (F P )B(H) (the Fuglede-Putnam property). Some open questions
are also given.
Let A, B ∈ B(H). We say that the pair (A, B) satisfies (F P )B(H) , if
AC = CB where C ∈ B(H) implies A∗ C = CB ∗ .
2
Main results
In the following we will denote the spectrum, and the approximate spectrum
of an operator A ∈ B(H) by σ(A), and σa (A) respectively.
Lemma 2.1 If the matrix operator
"
Q R
S T
#
S. Mecheri
3
defined on H ⊕ H is invertible, then the operator
S ∗ S + Q∗ Q
is invertible on H.
Proof. Since
S ∗ S + Q∗ Q
is a positive operator,
σ(S ∗ S + Q∗ Q) = σa (S ∗ S + Q∗ Q).
If we assume that S ∗ S + Q∗ Q is not invertible, then there exists a sequence
(xn )n ⊂ H such that
kxn k = 1, ∀n ≥ 1 and limk(S ∗ S + Q∗ Q)xn k = 0.
Consequently,
"
2
#
*"
#∗ "
#
+
Q R
Q R
Q R
lim (xn ⊕ 0) = lim
(xn ⊕ 0), (xn ⊕ 0)
n→∞ S
n→∞
T
S T
S T
= lim ((S ∗ S + Q∗ Q)xn , xn ) = 0.
n→∞
which contradicts our hypotheses and the proof is complete .
2
Theorem 2.1 Let N be a normal operator and let A be an operator in
B(H). If the pair (A, N ) (resp. (N, A) has the property (F P )B(H) , then the
equations
N X − XA = C( respect. AX − XN = C)
have a solution X if and only if
"
N 0
0 A
"
(resp.
#
N 0
0 A
are similar operators on H ⊕ H.
"
and
#
N C
0 A
"
and
#
#
N C
)
0 A
Why to solve the operator equation AX − XB = C
4
Proof. If the equation
N X − XA = C
has a solution X, then
"
I −X
0 I
#"
N 0
0 A
Therefore
#"
"
I X
0 I
#
N 0
0 A
#
N 0
0 A
#
"
N N X − XA
0 A
=
"
N C
0 A
#
"
N C
0 A
#
and
#
"
=
N C
0 A
are similar.
Conversely, if
"
and
are similar, then there exists an invertible matrix operator
"
Q R
S T
#
on B(H ⊕ H) such that
"
N 0
0 A
#"
Q R
S T
#
"
=
Q R
S T
#"
N C
0 A
#
.
Hence
QN = N Q, N R − RA = QC, AS = SN, AT − T A = SC.
By applying the property (F P )B(H) , we obtain
AS ∗ = S ∗ N and N Q∗ = Q∗ N.
Therefore
N S ∗ S = S ∗ SN,
that is, N commutes with S ∗ S and T ∗ T. Furthermore we have
(S ∗ S + Q∗ Q)C = Q∗ (N R − RA) + S ∗ (AT − T A)
= (N Q∗ R + N S ∗ T ) − (Q∗ RA + S ∗ T A)
#
.
S. Mecheri
5
= N (Q∗ R + S ∗ T ) − (Q∗ R + S ∗ T )A.
By Lemma 2.1 the operator
S ∗ S + Q∗ Q
is invertible and commute with N . Hence
N X − XA = C,
X = (S ∗ S + Q∗ Q)−1 (Q∗ R + S ∗ T ).
If the pair (N, A) satisfies the (F P )B(H) property, then the equation
AX − XN = C
has a solution X given by
X = −(QS ∗ + RT ∗ )(SS ∗ + T T ∗ )−1 .
2
Corollary 2.1 Let N, A be two operators in B(H) with A normal. If the
pair (A, N ) (resp. (N, A)) has the (F P )B(H) property, then
("
R(δA,N ) =
N 0
0 A
#
A 0
0 N
#
"
and
#
N C
0 A
)
are similar
respectively
("
R(δN,A ) =
"
and
A C
0 N
#
)
are similar ,
where δA,B is the generalized derivation defined on B(H) by
δA,B (X) = AX − XB
.
Theorem 2.2 Let ΓI be the collection of pairs of operators (A, B) satisfying the (F P )B(H) property. Then the following assertions are equivalent:
(i) (R, S) ∈ ΓI if R and S are unitary equivalent to A and B respectively.
(ii) (B ∗ , A∗ ) ∈ ΓI .
(iii) (A−1 , B −1 ) ∈ ΓI if A and B are invertible.
(iv) (λA, λB) ∈ ΓI for all λ ∈ C
I.
(v) (λI + A, λI + B) ∈ ΓI for all λ ∈ C
I.
Why to solve the operator equation AX − XB = C
6
Proof. i) Assume that R and S are unitary equivalent to A and B
respectively. Then there exist two unitary operators U and V such that
R = U AU ∗ and S = V BV ∗ .
If
RX = XS, f or X ∈ B(H)
then
AU ∗ XV = U ∗ XV B.
Now since (A, B) ∈ ΓB(H) , it results that U ∗ XV ∈ B(H). Therefore
A∗ U ∗ XV = U ∗ XV B ∗ .
By this we obtain
U A∗ U ∗ X = XV B ∗ V ∗ ,
from where
R∗ X = XS ∗ .
Which proves that (R, S) ∈ ΓI .
(ii) If B ∗ X = XA∗ for X ∈ B(H), then AX ∗ = X ∗ B and since X ∗ ∈
B(H),
A∗ X ∗ = X ∗ B ∗ ,
that is, XA = BX.
(iii) If
A−1 X = XB −1 , f or X ∈ B(H)
Then
A(A−1 X)B = A(XB −1 )B,
that is, AX = XB and so, A∗ X = XB ∗ . Hence
(A∗ )−1 A∗ X(B ∗ )−1 = (A∗ )−1 XB ∗ (B ∗ )−1 ,
therefore (A∗ )−1 X = X(B ∗ )−1 .
(iv) If
(λA)X = X(λB), f or X ∈ B(H),
then AX = XB and hence
A∗ X = XB ∗ .
S. Mecheri
7
Consequently
λA∗ X = XλB ∗ .
(v ) if
(A + λI)X = X(B + λI),
then AX = XB. Therefore
A∗ X = XB ∗
and hence
(A + λI)∗ X = X(B + λI)∗ .
The (F P )B(H)
corollary.
2
property hypothesis can be weakned as in the following
Corollary 2.2 Let ΩI be the collection of pairs of operators (A, N ), (N, A)
for which N X − XA = C (resp. AX - XN =C) have solutions X . Assume
that (A, N ) ∈ ΩI (resp. (N, A) ∈ ΩI ) then
(i) (R, S) ∈ ΩI (resp. (S, R) ∈ ΩI ) if R and S (resp. S and R are
unitary equivalent to A and N (resp. to N and A).
(ii) (N ∗ , A∗ ) ∈ ΩI (resp. (A∗ , N ∗ ) ∈ ΩI )
(iii) (A−1 , N −1 ) ∈ ΩI (resp. (A−1 , N −1 ) ∈ ΩI ) if A and N are invertible.
(iv) (λA, λN ) ∈ ΩI for all λ ∈ C
I (resp. (λN, λA) ∈ ΩI for all λ ∈ C
I ).
(v) (λI + A, λI + N ) ∈ ΩI for all λ ∈ C
I (resp.(λI + A, λI + N ) ∈ ΩI
for all λ ∈ C
I) .
For any operator A in B(H) set, as usual, [A∗ , A] = A∗ A − AA∗ (the
self commutator of A), and consider the following standard definitions: A is
hyponormal if if [A∗ , A] is nonnegative, normal if A∗ A = AA∗ , subnormal if
it admits a normal extension. An operator A ∈ B(H) is called dominant by
J.G.Stampfli and B.L.Wadhwa [15] [6] if, for all complex λ, range(A − λ) ⊆
range(A − λ)∗ , or equivalently, if there is a real number Mλ ≥ 1 such
that k(A − λ)∗ f k ≤ Mλ k(A − λ)f k , for all f ∈ H. If there exists a real
number M such that Mλ ≤ M for all λ, the dominant operator A is said
to be M -hyponormal. A 1-hyponormal is hyponormal. An operator A is
said to be p-hyponormal if (for some 0 < p ≤ 1 (A∗ A)2p ≤ (AA∗ )2p , kquasihyponormal if A∗k (A∗ A − AA∗ )Ak (k ∈ IN ). If k = 1, A is said to be
quasi-hyponormal.
8
Why to solve the operator equation AX − XB = C
Let (N ), (SN ), (H), (p − H) ,(D), Q(k) denote the classes constituting
of normal, subnormal, hyponormal, p-hyponormal operators , dominant ,
quasi-hyponormal and k-quasihyponormal operators. Then
(N ) ⊂ (SN ) ⊂ (H) ⊂ (m − H) ⊂ (D)
and
(N ) ⊂ (SN ) ⊂ (H) ⊂ (p − H)
Corollary 2.3 Let N be a normal operator and let A be an operator in
B(H). If the pair (A, N ) (resp. (N, A) has the property (F P )B(H) , then the
equations
N X − XA = C( respect. AX − XN = C)
have a solution X if and only if
"
N 0
0 A
"
(resp.
#
N 0
0 A
"
and
#
N C
0 A
"
and
#
#
N C
)
0 A
are similar operators on H ⊕ H under either of the following cases:
(i) A dominant.
(ii) A p-hyponormal.
(ii) A k- quasihyponormal.
Proof. It is well known [15], [3] that the pair (N, A) (resp.(A, N ) has
the (F P )B(H) property under either of the above cases.
2
S. Mecheri
3
9
Some Problems
The operator A ∈ B(H) is said to be finite [16] if ||I − (AX − XA)|| ≥ 1
(*) for all X ∈ B(H), where I is the identity operator. The well-known
inequality (*), due to [16] is the starting point of the topic of commutator
approximation (a Topic which has its roots in quantum theory [17]). This
topic deals with minimizing the distance, measured by some norm or other,
between a varying commutator (or self-commutator XX ∗ −X ∗ X) and some
fixed operator [1, 6, 8]
we begin by the definition of the best approximant of an operator. Let
E be a normed space and M a supspace of E. If to each A ∈ E there exists
an operator B ∈ M for which
kA − Bk ≤ kA − Ck f or all C ∈ M.
Such B (if they exist) are called best approximants to A from M . To
approach the concept of an approximant consider a set of mathematical
objects(complex numbers, matrices or linear operator, say) each of which
is, in some sense, ”nice”, i.e. has some nice property P (being real or selfadjoint, say): and let A be some given, not nice, mathematical object: then
a P best approximants of A is a nice mathematical object that is ”nearest
” to A. Equivalently, a best approximant minimizes the distance between
the set of nice mathematical objects and the given, not nice object.
Of course, the terms ” mathematical object”, ”nice”, ”nearest”, vary
from context to context. For a concrete example, let the set of mathematical
objects be the complex numbers, let ”nice”=real and let the distance be
measured by the modulus, then the real approximant of the complex number
. Thus for all real x
z is the real part of it, Rez = (z+z)
2
|z − Rez| ≤ |z − x|.
3.1
Problem I
The related topic of approximation by commutators AX-XA or by generalized commutator AX-XB, which has attracted much interest, has its
roots in quantum theory. The Heinsnberg Uncertainly principle may be
mathematically formulated as saying that there exists a pair A, X of linear
transformations and a non-zero scalar α for which
AX − XA = αI
(3.1)
Why to solve the operator equation AX − XB = C
10
Clearly, (3.1) cannot hold for square matrices A and X and for bounded
linear operators A and X. This prompts the question:
how close can AX − XA be the identity?
Williams [16] proved that if A is normal, then, for all X in B(H),
||I − (AX − XA)|| ≥ ||I||.
(3.2)
Mecheri [7] generalized Williams inequality (3.2): he proved that if A, B
are normal, then for all X ∈ B(H)
||I − (AX − XB)|| ≥ ||I||.
(3.3)
Anderson [1] generalized Williams inequality (3.2): he proved that if A
is normal and commutes with B then, for all X ∈ B(H)
||B − (AX − XA)|| ≥ ||B||
(3.4).
Maher [6] obtained the Cp variants of Anderson’s result. Mecheri [8] studied
approximation by generalized commutators AX-XC: he showed that the
following inequality holds
||B − (AX − XC)||p ≥ ||B||p
(3.5).
for all X ∈ Cp if and only if B ∈ kerδA,B
In the above inequalities (3.2),(3.3), (3.4) and (3.5) the zero commutator
is a commutator approximant in Cp of B.
3.2
Problem II
Let δA be the operator defined on B(H) by δA (X) = AX − XA. It is
known that I is not commutator,i.e. I 6∈ R(δA ). Anderson [1] proved that
there exists A ∈ B(H) such that I ∈ R(δA ), that is, the distance from I to
AX − XA is minimal, i.e., equal to zero. For more details see Mecheri[7]
In [8] We constructed a pair (A, X) of elements in B(H) such that
dist(I, R(δA )) < 1.
Open question: Does dist(I, R(δA )) = r ∈ (0, 1) implies for all invertible
S that dist(I, R(δSAS −1 )) = r ∈ (0, 1)
S. Mecheri
3.3
11
Problem III
Let
JA (H) = {A ∈ B(H) : I ∈ R(δA )}.
Here is a problem that might of interest. Recall [5] if T : X → Y define
T = {limn T xn : supn kxn k < ∞},
where X, Y are Banach spaces. differing from the usual closure in that its
points have to be the limits of images of bounded sequences of vectors so:
Question. For which operators A on Hilbert space H do we have
I ∈ R(δA ) ?
References
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normal derivation and related operators, Pacific J. Math., 61(1976)
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12
Why to solve the operator equation AX − XB = C
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