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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.)
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