A Fuglede-Putnam theorem modulo the Hilbert-Schmidt class for almost normal operators with finite modulus of Hilbert-Schmidt quasi-triangularity

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1 Concr. Oper. 2016; 3: 8 14 Concrete Operators Open Access Research Article Vasile Lauric* A Fuglede-Putnam theorem modulo the Hilbert-Schmidt class for almost normal operators with finite modulus of Hilbert-Schmidt quasi-triangularity DOI /conop Received June 7, 2015; accepted January 9, Abstract: We extend the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to almost normal operators with finite Hilbert-Schmidt modulus of quasi-triangularity. Keywords: Hilbert-Schmidt trace-class operators, Almost normal operators, Almost hyponormal operators, Modulus of quasi-triangularity MSC: 47B20 1 Introduction Im memory of my dear father, Atanasie. Let H be a separable, infinite dimensional, complex Hilbert space, denote by L.H/ the algebra of all bounded linear operators on H by C p.h/ the Schatten p-classes, with particular ones C 1.H/ C 2.H/; the traceclass the Hilbert-Schmidt class, respectively. For an operator T 2 L.H/; let D T denote its self-commutator ŒT ; T WD T T T T : An operator T for which D T 0; (D T 2 C 1 ; or.d T / 2 C 1.H/) is called hyponormal, (almost normal, or almost hyponormal, respectively), where A denotes the negative part, that is jaj A ; of a selfadjoint 2 operator A: We will denote these classes by H0 1.H/; AN.H/; H 1 1.H/; respectively. The classical Fuglede-Putnam theorem states that for normal operators M; N 2 L.H/ for an X 2 L.H/ such that MX XN D 0; implies M X XN D 0: Weiss (see [17, Theorem 1] extended this result by proving that if M; N 2 L.H/ are normal operators X is in L.H/ such that MX XN 2 C 2.H/, then M X XN 2 C 2.H/ jjm X XN jj 2 D jjmx XN jj 2 : Although it is not the goal of this note, we mention that the above estimate was extended (cf. [1, 8]) to arbitrary Schatten p-classes as follows. If N; X 2 L.H/; N is a normal operator p > 1; then jjn X XN jj p c p jjnx XN jj p : In [9, Corollary 8.6] was proved that there exist a normal operator N 2 L.H/ a compact operator X such that NX XN 2 C 1.H/ N X XN C 1.H/: In [2] [5] some extensions of Weiss result for subnormal operators were obtained (recall that a subnormal operator is the restriction of a normal operator to an invariant subspace) by proving that if M; N 2 L.H/ are *Corresponding Author: Vasile Lauric: Department of Mathematics, Florida A&M University, Tallahassee, FL 32307, USA, vasile.lauric@famu.edu 2016 Lauric, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

2 A Fuglede-Putnam theorem modulo the Hilbert-Schmidt class 9 subnormal operators X 2 L.H/ such that MX XN 2 C 2.H/; then M X XN 2 C 2.H/ jjm X XN jj 2 jjmx XN jj 2 : In [5] Kittaneh provided an example of subnormal operators M; N 2 L.H/ an operator X 2 L.H/ with MX XN 2 C 2.H/; but M X XN C 2.H/: Also he began to investigate Fuglede-Putnam generalizations for almost normal operators. For instance, Theorem 2.2 states that if S 2 L.H/ is an almost normal subnormal operator SX XS 2 C 2.H/ for some X 2 L.H/; then S X XS 2 C 2.H/; Theorem 2.4 provides a similar result for an almost normal operator T so that T 2 is normal. Along the same line of results, Nakazi [7] proved that if T 2 H0 1.H/ X 2 L.H/ such that TX D XT; then jjt X XT jj 2fArea..T //g 1 2 jjxjj: In [9] Shulman Turowska proved that for operators A; B definition below), then for each X 2 L.H/; 2 H0 1.H/ so that A has finite multiplicity (see jja X XB jj 2 jjax XBjj 2 : The goal of this note is to extend Fuglede-Putnam theorem to operators that are almost hyponormal operators also quasi-triangular modulo the Hilbert-Schmidt class. The interested reader can consult [4] for some applications of the Fuglede-Putnam theorem for normal operators. 2 Fuglede-Putnam type results Let P.H/ (or simply P) denote the set of all finite rank orthogonal projections on H. The modulus of quasitriangularity modulo C 2.H/ of an operator T 2 L.H/ (see [10], also [11 14] for further properties applications) is q 2.T / D lim inf jj.i P /TP jj 2 ; P 2P where lim inf is with respect to the natural order on P: The rational cyclic multiplicity (for short, multiplicity) of an operator T 2 L.H/; denoted by m.t /; is the smallest cardinal number m with the property that there are m vectors x 1 ; : : : ; x m 2 H such that _fr.t /x j j 1 j m; r 2 Rat..T //g D H; where Rat..T // denotes the algebra of complex-valued rational functions with poles off.t /; where.t / denotes the spectrum of T _ denotes the norm-closure of the linear span generated by the listed vectors. For an operator T 2 L.H/; according to Proposition 1 of [13], q 2.T /.m.t // 1 2 jjt jj : (1) It was observed by Hadwin-Nordgren (see [3], page ) that.t / m.t / where q 2.T /..T // 1 2 jjt jj.m.t // 1 2 jjt jj ; (2).T / D lim inf P 2P rank..i P /TP / : According to Proposition 1 of [14], if T 2 AN.H/ q 2.T / < 1; then q 2.T / < 1 there exists a sequence P n 2 P; n 1; such that P n " I strongly, lim jj.i P n/tp n jj 2 D q 2.T / lim jj.i P n/t P n jj 2 D q 2.T /: n1 n1 The quasi-triangularity modulo C 2.H/ was used by Voiculescu to give an extension of Berger-Shaw inequality with an elegant proof, depending only on operator theoretic concepts.

3 10 V. Lauric Theorem A ([13, Propositions 2, 3]). If T 2 H 1 1.H/; then Tr.D T /.q 2.T // 2 : Moreover, if X 2 C 2.H/ such that m.t C X/ < 1; then T 2 AN.H/ Tr.D T / m.t C K/ Area..T C K//: This result was extended (see Theorem 1 of [3]) to the case m.t C X/ D 1 but additional hypothesis that Area..T C X// D 0; with the conclusion that Tr.D T / D 0: We will use the following idea that was first used by Furuta [2], namely if T 1 ; T 2 2 L.H/; then the operator T1 ;T 2.X/ WD T 1 X XT 2 defined on the Hilbert space C 2.H/ (with inner product hx; Y i 2 D Tr.XY /) has its adjoint given by T 1 ;T 2.X/ D T1 X XT 2 the self-commutator of 12 WD T1 ;T 2 is D 12.X/ D D T1 X XD T2 D DT1 ;D T2.X/: Theorem 2.1. Let S 2 AN.H/ with q 2.S/ < 1 let X 2 L.H/ such that SX S X XS 2 C 2.H/: XS 2 C 2.H/: Then Proof. Denote SX XS by R S X XS by Q: Let P n 2 P; n 1; such that P n " I strongly lim jj.i P n/sp n jj 2 D q 2.S/ lim jj.i P n/s P n jj 2 D q 2.S / < 1: n1 n1 Then jjqp n jj 2 2 D Tr.P nq QP n / " Tr.Q Q/: We need to prove that the sequence fjjqp n jj 2 g n is bounded above. Write S D S 1n S 2n X D X 1n X 2n ; S 3n S 4n X 3n X 4n relative to the decomposition of H as P n H.I P n /H: Thus Q D S 1n X 1n C S3n X 3n X 1n S1n X 2n S2n S2n X 1n C S4n X 3n X 3n S1n X 4n S2n D Q 11 C Q 32 ; Q 41 C Q 22 where Q 11 D S 1n X 1n X 1n S 1n ; Q 32 D S 3n X 3n X 2n S 2n ; Q 41 D S 4n X 3n X 3n S 1n ; Q 22 D S 2n X 1n X 4n S 2n. Consequently, jjqp njj 2 2 D jjq 11 C Q 32 jj 2 2 C jjq 41 C Q22jj 2 2 : Next we estimate the Hilbert-Schmidt norm of each of the Q s. We begin with Q 32 ; namely jjq 32 jj 2 2 D jjs 3n X 3n X 2n S 2n jj2 2.jjXjj jjs 3n jj 2 C jjxjj jjs 2n jj 2/ 2 2 jjxjj 2.jjS 3n jj2 2 C jjs 2n jj2 2 / / 2 jjxjj2 q 2 2.S/ C q2 2.S / ; jjq 32 jj 2 2 / 2 jjxjj2 q 2 2.S/ C q2 2.S / ; (3) where the symbol / used above in what follows means that the terms jjs 3n jj 2 2 jjs 2njj 2 2 to the right h side of / were replaced already by their limit q2 2.S/ or q2 2.S /; respectively. Denote 11.X/ D S 1n X XS 1n defined on C 2.P n H/I therefore Q 11 D 11.X 1n/ D S 1n X 1n X 1n S 1n ; Œ 11 ; 11.X 1n / D D S1n X 1n X 1n D S1n ; then jj 11.X 1n /jj 2 2 jj 11.X 1n/jj 2 2 D hœ 11 ; 11.X 1n /; X 1n i 2 D D Tr.D S1n X 1n X 1n / A matrix calculation shows that P n D S P n D D S1n C S 3n S 3n Tr.X 1nD S1n X 1n / D D Tr.X 1n D S 1n X 1n / Tr.X 1n D S1n X 1n /: S 2n S2n ; X 1n D S 1n X 1n D X 1n D SX 1n X 1n S 3n S 3nX 1n C X 1n S 2nS 2n X 1n

4 A Fuglede-Putnam theorem modulo the Hilbert-Schmidt class 11 X 1n D S1n X 1n D X 1nD S X 1n X 1n S 3n S 3nX 1n C X 1nS 2n S 2n X 1n ; respectively. Therefore jj 11.X 1n/jj 2 2 jj 11.X 1n /jj 2 2 D Tr.X 1n D S 1n X 1n / C Tr.X 1n D S1n X1n / D Tr.X 1n D S X1n / Tr.X 1nS3n S 3nX1n / C Tr.X 1nS 2n S2n X 1n / Tr.X1n D SX 1n / C Tr.X1n S 3n S 3nX 1n / Tr.X1n S 2nS2n X 1n/ Tr.X 1n D S X1n / Tr.X 1n D SX 1n /C C Tr.X 1n S 2n S2n X 1n / C Tr.X 1n S 3n S 3nX 1n / 2 jjxjj 2 jjd S jj 1 C jjs2n X 1n jj2 2 C jjs 3nX 1n jj 2 2 / / jjxjj 2 2 jjd S jj 1 C q2 2.S/ C q2 2.S / ; consequently jjq 11 jj 2 2 / jjr 11jj 2 2 C jjxjj2 2 jjd S jj 1 C q 2 2.S/ C q2 2.S / ; (4) where R 11 D S 1n X 1n X 1n S 1n D 11.X 1n /: Combining (3) (4), jjq 11 C Q 32 jj jjQ 11jj 2 2 C jjq 32jj 2 2 / / / 2 jjr 11 jj 2 2 C jjxjj2 4 jjd S jj 1 C 6 q 2 2.S/ C 6 q2 2.S / : (5) The operator Q 41 D S 4n X 3n X 3n S1n can be viewed as 41.X 3n/; where 41 is the adjoint of 41 W C 2.P n H;.I P n /H/ C 2.P n H;.I P n /H/ defined by 41.Y / D S 4n Y YS 1n : Denoting 41.X 3n / by R 41 ; we have jjr 41 jj 2 2 jjq 41 jj 2 2 D jj 41.X 3n /jj 2 2 jj 41.X 3n/jj 2 2 D D hœ 41 ; 41.X 3n /; X 3n i 2 D Tr.D S4n X 3n X 3n / D Tr.X 3n D S 4n X 3n / Tr.X 3n D S1n X 3n /: Tr.X 3n D S1n X 3n / D Since X 3n D.I P n /XP n ; X3n D P nx.i P n /; using again the matrix representation of D S relative to the decomposition of H into P n H.I P n /H; that is P n D S P n D D S1n C S3n S 3n S 2n S2n ; we obtain X 3n D S1n X3n D X 3n.D S S3n S 3n C S 2n S2n /X 3n D D X 3n D S X 3n X 3n S 3n S 3nX 3n C X 3nS 2n S 2n X 3n : On other h.i P n /D S.I P n / D D S4n C S2n S 2n S 3n S3n ; which implies X 3n D S 4n X 3n D X 3n.D S S 2n S 2n C S 3n S 3n /X 3n D D X 3n D SX 3n X 3n S 2n S 2nX 3n C X 3n S 3nS 3n X 3n: Thus so jjq 41 jj 2 2 jjr 41 jj 2 2 D Tr.X 3n D S 4n X 3n / C Tr.X 3n D S1n X3n / D D Tr.X 3n D S X3n / Tr.X 3nS3n S 3nX3n / C Tr.X 3nS 2n S2n X 3n /C Tr.X3n D SX 3n / C Tr.X3n S 2n S 2nX 3n / Tr.X3n S 3nS3n X 3n/ Tr.X 3n D S X3n / Tr.X 3n D SX 3n /C C Tr.X 3n S 2n S2n X 3n / C Tr.X 3n S 2n S 2nX 3n / 2 jjxjj 2 jjd S jj 1 C jjs2n X 3n jj2 2 C jjs 2nX 3n jj 2 2 / / jjxjj 2 2 jjd S jj 1 C 2 q2 2.S / ; jjq 41 jj 2 2 / jjr 41jj 2 2 C jjxjj2 2 jjd S jj 1 C 2 q2 2.S / : (6)

5 12 V. Lauric Finally, Q 22 D S 2n X 1n X 4n S 2n can be hled similarly to Q 32; namely jjq 22 jj 2 jjs 2n jj 2 jjx 1njj C jjx 4n jj jjs 2n jj 2 2 jjxjj jjs 2njj 2 ; jjq 22 jj 2 2 / 4 jjxjj2 q 2 2.S /: (7) According to (6) (7), consequently, jjq 41 C Q 22 jj jjQ 41jj 2 2 C jjq 22jj 2 2 / / 2 jjr 41jj 2 2 C jjxjj2 4 jjd S jj 1 C 12 q 2 2.S // ; (8) jjqp n jj 2 2 / 2.jjR 11jj 2 2 C jjr 41jj 2 2 / C jjxjj2 8 jjd S jj 1 C 6 q 2 2.S/ C 18 q2 2.S //: (9) The proof will be finished after we establish that jjr 11 jj 2 2 C jjr 41jj 2 2 is bounded. Since R 2 C 2.H/, we have jjrp n jj 2 2 " Tr.R R/, jjrp n jj 2 2 is bounded above by jjrjj2 2. The representation of R relative to the decomposition of H into P n H.I P n /H is R 11 C R 23 ; R 41 C R 33 where R 23 D S 2n X 3n X 2n S 3n R 33 D S 3n X 1n X 4n S 3n ; therefore On other h, jjrp n jj 2 2 D jjr 11 C R 23 jj 2 2 C jjr 41 C R 33 jj 2 2 : (10) jjr 11 jj 2 2.jjR 11 C R 23 jj 2 C jjr 23 jj 2 / 2 2.jjR 11 C R 23 jj 2 2 C jjr 23jj 2 2 / Similarly therefore jjr 23 jj 2 2 / 2 jjxjj2 q 2 2.S/ C q2 2.S / ; jjr 11 jj 2 2 / 2 jjr 11 C R 23 jj 2 2 C 4 jjxjj2 q 2 2.S/ C q2 2.S / : (11) jjr 41 jj 2 2.jjR 41 C R 33 jj 2 C jjr 33 jj 2 / 2 2.jjR 41 C R 33 jj 2 2 C jjr 33jj 2 2 / jjr 33 jj 2 2 / 4 jjxjj2 q 2 2.S/; jjr 41 jj 2 2 / 2.jjR 41 C R 33 jj 2 2 C 8 jjxjj2 q2 2.S//: (12) According to (11) (12), jjr 11 jj 2 2 C jjr 41jj 2 2 / 2.jjR 11 C R 23 jj 2 C jjr 41 C R 33 jj 2 2 / C jjxjj2.12 q 2 2.S/ C 4 q2 2.S //; (13) jjr 11 jj 2 2 C jjr 41jj 2 2 / 2jjRP njj 2 2 C jjxjj2 12 q 2 2.S/ C 4 q2 2.S / : (14) Finally, according to (9) (14), jjqp n jj 2 2 / 4 jjrp njj 2 2 C jjxjj2 8 jjd S jj 1 C 30 q 2 2.S/ C 26 q2 2.S / ; jjqjj jjrjj2 2 C jjxjj2 8 jjd S jj 1 C 30 q 2 2.S/ C 26 q2 2.S / ; which ends the proof.

6 A Fuglede-Putnam theorem modulo the Hilbert-Schmidt class 13 Theorem 2.2. Let S 1 ; S 2 2 AN.H/ with q 2.S 1 / q 2.S 2 / < 1 let X 2 L.H/ such that S 1 X XS 2 2 C 2.H/: Then S 1 X XS 2 2 C 2.H/: Proof. Let S D S 1 S 2 Y D SY YS D 0 S 1X XS 2 0 X : Thus S 2 AN.H H/; q2 2.S/ q2 2.S 1/ C q2 2.S 2/ < 1; 2 C 2.H H/: According to Theorem 2.1, S Y YS D 0 S 1 X XS 2 2 C 2.H H/; S 1 X XS 2 is in C 2.H/: Corollary 2.3. Let S 2 AN.H/ with q 2.S/ < 1 let X 2 L.H/ such that SX S X XS 2 C 2.H/: XS 2 C 2.H/: Then Proof. According to [14, Proposition 1], for S 2 AN.H/ with q 2.S/ < 1; the following holds: q 2.S / < 1 Theorem 2.2 can be applied. q 2 2.S/ D q2 2.S / C Tr.D S /; Theorem 2.4. Let S 1, S2 2 H 1 1.H/ with q 2.S 1 / q 2.S2 / < 1 let X 2 L.H/ such that S 1X XS 2 belongs to C 2.H/. Then S1 X XS 2 belongs to C 2.H/: Proof. Let S 1 ; S2 2 H 1 1.H/ put W D S 1 S2 Y D 0 X I thus W 2 H1 1.H H/; q2 2.W / q2 2.S 1/ C q 2.S2 / < 1; according to Theorem A, W 2 AN.H H/ Applying Corollary 2.3, one gets W Y Y W D 0 S 1X XS 2 W Y Y W D 0 S 1 X XS 2 2 C 2.H H/: 2 C 2.H H/; S 1 X XS 2 2 C 2.H/: An application of Theorem A Hadwin-Nordgren case when area of the spectrum is zero leads to the following. Corollary 2.5. If for S 1 ; S2 2 H 1 1.H/ there are K i 2 C 2.H/; i D 1; 2; so that both m.s 1 CK 1 / m.s2 CK 2/ are finite or both.s 1 C K 1 /.S 2 C K 2 / have area zero if X 2 L.H/ is such that S 1 X XS 2 2 C 2.H/; then S1 X XS 2 2 C 2.H/: We close this note with the remark that an affirmative answer to Voiculescu s Conjecture 4 (see [15] or [16]) would imply the following. Question (G). Is is true that if S 2 AN.H/ X 2 L.H/ such that SX XS 2 C 2.H/; then S X XS 2 C 2.H/? Recall that Voiculescu s Conjecture 4 states that if S 2 AN.H/; then there exists T 2 AN.H/ such that S T is the sum of a normal operator a Hilbert-Schmidt operator. Indeed, if for S 2 AN.H/; there exist T 2 AN.H/;

7 14 V. Lauric a normal operator N; a Hilbert-Schmidt operator K such that S T D N C K; then.s T / XQ a Hilbert-Schmidt operator, where XQ D X 0; then according to Weiss theorem ([17, Theorem 1]), Q X.S T / is.s T / Q X Q X.S T / 2 C 2.H H/; consequently S X XS 2 C 2.H/: If (G) is true, then by a stard argument, it can be extended to two almost normal operators, that is, if S; T 2 AN.H/ X 2 L.H/ such that SX XT 2 C 2.H/; then S X XT 2 C 2.H/: References [1] A. Abdessemed E. B. Davies, Some commutator estimates in the Schatten classes, J. London Math. Soc. (2), 41, 1989, [2] T. Furuta, An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality, Proc. Amer. Math. Soc., 81, 1981, [3] D. Hadwin E. Nordgren, Extensions of the Berger-Shaw theorem, Proc. Amer. Math. Soc., 102, 1988, [4] E. Kissin, D. Potapov, V. Shulman F. Sukochev, Operator smoothness in Schatten norms for functions of several variables: Lipschitz conditions, differentiability unbounded derivations, Proc. London Math. Soc. (4), 105, 2012, [5] F. Kittaneh, On generalized Fuglede-Putnam theorems of Hilbert-Schmidt type, Proc. Amer. Math. Soc., 88, 1983, [6] F. Kittaneh, On Lipschitz functions of normal operators, Proc. Amer. Math. Soc., 94, 1985, [7] T. Nakazi, Complete spectral area estimates self-commutators, Michigan Math. J., 35, 1988, [8] V. Shulman, Some remarks on the Fuglede-Weiss theorem, Bull. London Math. Soc. (4), 28, 1996, [9] V. Shulman L. Turowska, Operator Synthesis II. Individual synthesis linear operator equations, J. für die reine und angewte Math. (590), 2006, 2006, [10] D. Voiculescu, Some extensions of quasitriangularity, Rev. Roumaine Math. Pures Appl., 18, 1973, [11] D. Voiculescu, Some results on norm-ideal perturbation of Hilbert space operators, J. Operator Theory, 2, 1979, 3 37 [12] D. Voiculescu, Some results on norm-ideal perturbation of Hilbert space operators II, J. Operator Theory, 5, 1981, [13] D. Voiculescu, A note on quasitriangularity trace-class self-commutators, Acta Sci. Math. (Szeged), 42, 1980, [14] D. Voiculescu, Remarks on Hilbert-Schmidt perturbations of almost normal operators, Topics in Modern Operator Theory; Operator Theory: Advances Applications-Birkhäuser, 2, 1981, [15] D. Voiculescu, Almost Normal Operators mod Hilbert-Schmidt the K-theory of the Algebras Eƒ./, arxiv: v2 [math.oa], 2011 [16] D. Voiculescu, Hilbert space operators modulo normed ideals, Proc. Int. Congress Math., 1983, [17] G. Weiss, The Fuglede commutativity theorem modulo operator ideals, Proc. Amer. math. Soc., 83, 1981, [18] G. Weiss, Fuglede s commutativity theorem modulo the Hilbert-Schmidt class generating functions for matrix operators. II, J. Operator Theory, 5, 1981, 3 16