关于Hilbert空间上算子乘积有限和的奇异值不等式

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Singular Value Inequalities for Finite Sum of Products of Operators on Hilbert Space

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Abstract:

Let $A_k,B_k,X_{kl}\ (k=1,2,\cdots,n,\mbox{ and }l=1,2,\cdots,n)$ be bounded linear operators on a complex separable Hilbert space ${\mathcal H}.$ In the present paper, it is shown that if $X_{kl}\ (k,l=1,2,\cdots,n)$ are compact, then

$s_j\Big(\sum\limits_{k,l=1}^nA_kX_{kl}B_l\Big)\leqslant \sqrt{\Big\|\sum\limits_{k=1}^n|A_k^*|^2\Big\|}\cdot\sqrt{\Big\|\sum\limits_{l=1}^n|B_k|^2\Big\|} s_j(( X_{kl})_{n\times n})$
for

$j=1,2,\cdots,$ where

$\|\cdot\|$ is the usual operator norm and

$( X_{kl})_{n\times n}$ is the operator defined on

$\oplus_{k=1}^n{\mathcal H}$ by $( X_{kl})_{n\times n} =\left(\begin{array}{ccc}X_{11}&~~\cdots~~ &X_{1n}\\\vdots&\ddots&\vdots\\X_{n1}&\cdots &X_{nn}\end{array}\right).

Key words: Singular value, Products of operators, Compact operator, Inequality

CLC Number:

47A30

FANG Li, ZHANG Hai-Yan. Singular Value Inequalities for Finite Sum of Products of Operators on Hilbert Space[J].Acta mathematica scientia,Series A, 2011, 31(5): 1385-1392.

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