关键词: 奇异值, 算子乘积, 紧算子, 不等式

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