×

zbMATH — the first resource for mathematics

Singular values, norms, and commutators. (English) Zbl 1188.47018
Let \(A_i, B_i, X_i\,(1\leq i\leq n)\) be bounded linear operators on a separable Hilbert space such that all \(X_i\)’s are compact. The authors extend a result of O. Hirzallah [Linear Algebra Appl. 431, No. 9, 1571–1578 (2009; Zbl 1172.47025)] by proving that the singular values of \(\sum_{i=1}^n A_iX_iB_i\) are dominated by those of \(\left(\sum_{i=1}^n \|A_i\|\,\|B_i\|\right) \left(\oplus_{i=1}^n X_i\right)\), where \(\|\cdot\|\) denotes the usual operator norm. They show that, if \(A\) and \(B\) are selfadjoint operators such that \(a_1 \leq A \leq a_2\) and \(b_1 \leq B \leq b_2\) for some real numbers \(a_1, a_2, b_1, b_2\) and if \(X\) is compact, then the singular values of the generalized commutator \(AX - XB\) are dominated by those of \(\max\{b_2-a_1, a_2-b_1\}(X \oplus X)\). This inequality proves a recent conjecture by F. Kittaneh [Linear Algebra Appl. 430, No. 8–9, 2362–2367 (2009; Zbl 1162.47010)]

MSC:
47A63 Linear operator inequalities
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B47 Commutators, derivations, elementary operators, etc.
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bhatia, R., Matrix analysis, (1997), Springer-Verlag New York
[2] Bhatia, R.; Kittaneh, F., Commutators, pinchings, and spectral variation, Oper. matrices, 2, 143-151, (2008) · Zbl 1147.15019
[3] Gohberg, I.C.; Krein, M.G., Introduction to the theory of linear nonselfadjoint operators, (1969), Amer. Math. Soc. Providence, RI · Zbl 0181.13504
[4] Hirzallah, O., Commutator inequalities for Hilbert space operators, Linear algebra appl., 431, 1571-1578, (2009) · Zbl 1172.47025
[5] Kittaneh, F., A note on the arithmetic – geometric Mean inequality for matrices, Linear algebra appl., 171, 1-8, (1992) · Zbl 0755.15008
[6] Kittaneh, F., Inequalities for commutators of positive operators, J. funct. anal., 250, 132-143, (2007) · Zbl 1131.47009
[7] Kittaneh, F., Norm inequalities for commutators of positive operators and applications, Math. Z., 258, 845-849, (2008) · Zbl 1139.47009
[8] Kittaneh, F., Norm inequalities for commutators of normal operators, (), 147-154 · Zbl 1266.47037
[9] Kittaneh, F., Norm inequalities for commutators of self-adjoint operators, Integral equations operator theory, 62, 129-135, (2008) · Zbl 1195.47008
[10] Kittaneh, F., Singular value inequalities for commutators of Hilbert space operators, Linear algebra appl., 430, 2362-2367, (2009) · Zbl 1162.47010
[11] Niezgoda, M., Commutators and accretive operators, Linear algebra appl., 431, 1192-1198, (2009) · Zbl 1171.47031
[12] Wang, Y.-Q.; Du, H.-K., Norms for commutators of self-adjoint operators, J. math. anal. appl., 342, 747-751, (2008) · Zbl 1139.47010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.