Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2008; 45(3): 771-780

Printed May 1, 2008

Copyright © The Korean Mathematical Society.

Weyl spectrum of the products of operators

Xiaohong Cao

Shaanxi Normal University

Abstract

Let $M_C=\left(\begin{smallmatrix}A&C\\0&B\\\end{smallmatrix}\right)$ be a $2\times 2$ upper triangular operator matrix acting on the Hilbert space $H\oplus K$ and let $\sigma_w(\cdot)$ denote the Weyl spectrum. We give the necessary and sufficient conditions for operators $A$ and $B$ which $\sigma_w\left(\begin{smallmatrix}A&C\\0&B\\\end{smallmatrix}\right)=\sigma_w\left(\begin{smallmatrix}A&0\\0&B\\\end{smallmatrix}\right)$ or $\sigma_w\left(\begin{smallmatrix}A&C\\0&B\\\end{smallmatrix}\right)=\sigma_w(A)\cup\sigma_w(B)$ holds for every $C\in B(K,H)$. We also study the Weyl's theorem for operator matrices.

Keywords: Weyl spectrum, Weyl's theorem, Browder's theorem, essential approximate point spectrum

MSC numbers: 47A05, 47A10, 47A55