Why we solve the operator equation AX − X B = C ∗†‡ Salah Mecheri 1 Abstract This work studies how certain problems in quantum theory have motivated some recent reseach in pure Mathematics in matrix and operator operator theory theory.. The mathema mathematic tical al key is that of a com commu mutat tator or or a generalized commutator, that is, find an operator X ∈ B(H ) satisfying the operator equation AX − X B = C . By this this we we will will show how and why to solve the operator equation AX − − X B = C . Some problems are studied and some open questions are also given.
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Introd trodu uction tion
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 commutat commutator. or. Given Given A, B ∈ B (H ). ) . The operat operator or C is said to be a commutator, if there exists an operator X ∈ B (H ) such that AX −X A = C . In general, if there exists an operator X ∈ B (H ) such that AX − X B = said to be a genera generali lized zed commuta commutator tor.. The first importa important nt C , then C is said contribution to the study of commutators is due to A. Wintner who in 1947 ∗
Key words: 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] †
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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 (1.1) I = AX − XA (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 = C B where C ∈ B (H ) implies A∗ C = C B ∗ .
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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
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S. Mecheri
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) = σ a(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
xn = 1, ∀n ≥ 1 and lim(S ∗ S + Q∗ Q)xn = 0. Consequently, lim
n→∞
Q R S T
2
(xn ⊕ 0)
= lim
n→∞
Q R S T
∗
Q R S T
(xn ⊕ 0), (xn ⊕ 0)
= lim ((S ∗ S + Q∗ Q)xn , xn ) = 0. n→∞
which contradicts our hypotheses and the proof is complete .
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
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Proof. If the equation
NX − XA = C
has a solution X , then
I −X 0 I
N 0 0 A
Therefore
I X = 0 I
N N X − XA = 0 A
N 0 0 A
N 0 0 A
and
N C 0 A
N C 0 A
N C 0 A
are similar. Conversely, if
and
are similar, then there exists an invertible matrix operator
Q R S T
Q R S T
on B (H ⊕ H ) such that
N 0 0 A
Q R = S T
N C 0 A
.
Hence QN = NQ, NR − RA = QC, AS = SN, AT − T A = S C.
By applying the property (F P )B(H ) , we obtain AS ∗ = S ∗ N andNQ∗ = 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 ∗ (NR − RA) + S ∗ (AT − T A) = (NQ∗ R + NS ∗ T ) − (Q∗ RA + S ∗ T A)
.
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S. Mecheri
= 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 .
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(δ A,N ) =
N 0 0 A
A 0 0 N
and
N C are similar 0 A
respectively R(δ N,A ) =
and
A C are similar , 0 N
where δ A,B is the generalized derivation defined on B (H ) by δ A,B (X ) = AX − XB
. Theorem 2.2 Let Γ I 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 ) ∈ Γ I if R and S are unitary equivalent to A and B respectively. (ii) (B ∗ , A∗) ∈ Γ I . (iii) (A−1 , B −1) ∈ Γ I if A and B are invertible. C . (iv) (λA, λB) ∈ Γ I for all λ ∈ I (v) (λI + A,λI + B ) ∈ Γ I for all λ ∈ I C .
Why to solve the operator equation AX − XB = C
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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 ∗ andS = V BV ∗ .
If RX = XS, for X ∈ B (H )
then AU ∗ XV = U ∗ X V B .
Now since (A, B ) ∈ Γ B(H ) , it results that U ∗ XV ∈ B (H ). Therefore A∗ U ∗ XV = U ∗ XV B ∗ .
By this we obtain U A∗ U ∗ X = X V B ∗ V ∗ ,
from where R∗X = X S ∗ .
Which proves that (R, S ) ∈ Γ I . (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 = X B −1 , for X ∈ B (H )
Then A(A−1 X )B = A (XB −1 )B,
that is, AX = X B and so, A∗X = X B ∗ . Hence (A∗)−1 A∗ X (B ∗ )−1 = (A∗)−1 XB ∗ (B ∗)−1 , therefore (A∗ )−1X = X (B ∗)−1 . (iv) If (λA)X = X (λB ), f or X ∈ B (H ), then AX = X B and hence A∗ X = X B ∗ .
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S. Mecheri
Consequently λA∗ X = X λB ∗ .
(v ) if (A + λI )X = X (B + λI ), then AX = X B . Therefore A∗ X = X B ∗
and hence (A + λI )∗ X = X (B + λI )∗.
The (F P )B (H ) property hypothesis can be weakned as in the following corollary. Corollary 2.2 Let Ω I be the collection of pairs of operators ( A, N ), ( N, A) for which NX − XA = C (resp. AX - XN =C) have solutions X . Assume that (A, N ) ∈ Ω I (resp. (N, A) ∈ Ω I ) then (i) (R, S ) ∈ Ω I (resp. (S, R) ∈ Ω I ) if R and S (resp. S and R are unitary equivalent to A and N (resp. to N and A). (ii) (N ∗ , A∗ ) ∈ Ω I (resp. (A∗ , N ∗ ) ∈ Ω I ) (iii) (A−1 , N −1 ) ∈ Ω I (resp. (A−1 , N −1 ) ∈ Ω I ) if A and N are invert-
ible. (iv) (λA, λN ) ∈ Ω I for all λ ∈ I C (resp. (λN,λA) ∈ Ω I for all λ ∈ I C ). I (resp.(λI + A,λI + N ) ∈ Ω I (v) (λI + A,λI + N ) ∈ Ω I for all λ ∈ C for all λ ∈ I C ) . 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 M λ ≥ 1 such that (A − λ)∗ f ≤ M λ (A − λ)f , for all f ∈ H. If there exists a real number M such that M λ ≤ 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)2 p ≤ (AA∗ )2 p , kquasihyponormal if A∗k (A∗A − AA∗ )Ak (k ∈ IN ). If k = 1, A is said to be quasi-hyponormal.
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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.
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S. Mecheri
3
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 X X ∗ − 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
A − B ≤ A − C forallC ∈ 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 b etween 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 z is the real part of it, Rez = (z+2 z ) . Thus for all real x
|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
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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 C p 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 ∈ C p if and only if B ∈ kerδ A,B In the above inequalities (3.2),(3.3), (3.4) and (3.5) the zero commutator is a commutator approximant in C p of B .
3.2
Problem II
Let δ A be the operator defined on B (H ) by δ A(X ) = AX − X A. It is R(δ A ). Anderson [1] proved that known that I is not commutator,i.e. I ∈ there exists A ∈ B (H ) such that I ∈ R (δ A ), 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(δ A )) < 1 .
Open question: Does dist(I, R(δ A )) = r ∈ (0 , 1) implies for all invertible S that dist(I, R(δ SAS − )) = r ∈ (0 , 1) 1
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S. Mecheri
3.3
Problem III
Let J A (H ) = { A ∈ B (H ) : I ∈ R (δ A )}.
Here is a problem that might of interest. Recall [5] if T : X → Y define T = { limnT xn : sup nxn < ∞} ,
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 (δ A ) ?
References [1] J.H.Anderson , C.Foias, Properties which normal operator share with normal derivation and related operators, Pacific J. Math., 61(1976) 313-325. [2] A. Bachir and A. Sagres, A. Generalized Fuglede-Putnam theorem and orthogonality. Aust. J. Math. Anal. Appl.1 (2004), no. 1, Art. 12, 5 pp. (electronic). [3] H.J. Chouan On the generalized quasi-hyponormal operators, J.Math. Wuhan., 5(1985), 23-32. [4] H.Flanders and H.K.Wimmer, On the matrix equations AX − XB = C and AX − Y B = C , SIAMJ.Appl.Math., 32(1977), 707-710. [5] H. Robin; L.W. Young, On the bounded closure of the range of an operator. Proc. Amer. Math. Soc. 125 (1997), no. 8, 2313–2318. [6] P.J. Maher, Commutator Approximants, Proc. Amer. Math. Soc., 115(1992), 995-1000. [7] S.Mecheri, Finite operators, Demonstratio Mathematica, 37(2002),357366 [8] S. Mecheri, Another version of Maher’s inequality, Zeitschrift fr Analysis und ihre Anwendungen., 23 (2004), no. 2, 303-311.
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Why to solve the operator equation AX − XB = C
[9] S.Mecheri, Global minimum and orthogonality in C p -classes., Math.Nachr, to appear [10] S. Mecheri, On minimizing S − (AX − XB ) p , Serdica Math. J. 26 (2000)., no. 2, 119-126. [11] S. Mecheri, Global minimum and orthogonality in C 1 -classes, J. Math. Anal. App., 287(2003) 51-60. [12] M.Rosenblum, On the operator equation AX − XB = Q , Duke. Math J., 23(1956), 263-269). [13] M.Rosenblum, On the operator equation AX − X B = Q with selfadjoint A, B , Proc.Amer.Math.Soc., 20(1969), 115-120. [14] W.E. Roth, The equations AX − Y B and AX − XB in matrices, Proc.Amer.Math.Soc., 3(1952), 392-316. [15] J.G.Stampfli and B.L. Wadhwa. An asymmetric Putnam-Fuglede theorem for dominant operators, Indiana Univ.Math.J.25(1976)., 359-365. [16] J.P.Williams., Finite operators., Proc. Amer. Math. Soc., 129-135, 26(1970). [17] H. Wielandt, ber die Unbeschrnktheit der Operatoren der Quantenmechanik. (German) Math. Ann. 121, (1949),21.
