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Some inequalities for commutators and an application to spectral variation. (English) Zbl 0752.47005
Let $$A,B$$ be two Hermitian or two unitary operators on a complex Hilbert space and let $$\Gamma\geq\gamma I$$ ($$\gamma>0$$) be a positive operator. The authors prove that for every unitarily invariant norm one has $$||| A\Gamma-\Gamma B|||>\gamma||| A- B|||$$.
It is known that when $$|||\cdot|||=\|\cdot\|_ 2$$, the Hilbert- Schmidt norm, this inequality remains true for any two normal operators $$A,B$$. The authors conjecture that the inequality is true in general for normal $$A,B$$. They also provide interesting corollaries, remarks and examples.

##### MSC:
 47A63 Linear operator inequalities 47B47 Commutators, derivations, elementary operators, etc. 15A42 Inequalities involving eigenvalues and eigenvectors 15B57 Hermitian, skew-Hermitian, and related matrices 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 65F99 Numerical linear algebra
##### Keywords:
unitarily invariant norm; Hilbert-Schmidt norm
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##### References:
 [1] Abdessemed, A. andDavies, E. B.,Some commutator estimates in the Schatten classes. J. London Math. Soc.39 (1989), 299–308. · Zbl 0692.47009 · doi:10.1112/jlms/s2-39.2.299 [2] Bhatia, R.,Perturbation bounds for matrix eigenvalues. Longman, Essex and Wiley, New York, 1987. · Zbl 0696.15013 [3] Bhatia, R.,Some inequalities for norm ideals. Commun. Math. Phys.111 (1987), 33–39. · Zbl 0632.47005 · doi:10.1007/BF01239013 [4] Bhatia, R., Davis, Ch. andKoosis, P.,An extremal problem in Fourier analysis with applications to operator theory. J. Funct. Anal.82 (1989), 138–150. · Zbl 0674.42002 · doi:10.1016/0022-1236(89)90095-5 [5] Bhatia, R., Davis, Ch. andMcIntosh, A.,Perturbation of spectral subspaces and solution of linear operator equations. Linear Algebra Appl.52–53 (1983), 45–67. [6] Bhatia, R. andKittaneh, F.,Norm inequalities for partitioned operators and an application. preprint. [7] Gohberg, I. C. andKrein, M. G.,Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, RI, 1969. [8] Hoffman, A. J. andWielandt, H. W.,The variation of the spectrum of a normal matrix. Duke Math. J.,20 (1953), 37–39. · Zbl 0051.00903 · doi:10.1215/S0012-7094-53-02004-3 [9] Kittaneh, F.,On generalized Fuglede-Putnam theorems of Hilbert-Schmidt type. Proc. Amer. Math. Soc.88 (1983), 293–298. · Zbl 0521.47014 · doi:10.1090/S0002-9939-1983-0695261-8 [10] Kittaneh, F.,On the continuity of the absolute value map in the Schatten classes. Linear Algebra Appl.118 (1989), 61–68. · Zbl 0676.47007 · doi:10.1016/0024-3795(89)90571-5 [11] Sun, J. G.,On the perturbation of the eigenvalues of a normal matrix (Chinese). Math. Numer. Sinica6 (1984), 334–336. · Zbl 0554.15009 [12] van Hemmen, J. L. andAndo, T.,An inequality for trace ideals. Commun. Math. Phys.76 (1980), 143–148. · Zbl 0449.47036 · doi:10.1007/BF01212822 [13] Weiss, G.,The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators, I, II. Trans. Amer. Math. Soc.246 (1978), 193–209, J. Operator Theory5 (1981), 3–16. · Zbl 0403.47009
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