WEYL SPECTRUM OF THE PRODUCTS OF OPERATORS
Xiaohong Cao · Journal of the Korean Mathematical Society · 2008
Let $M_C=\(\array{A & C\\0 & B}\)$ be a $2{\times}2$ upper triangular operator matrix acting on the Hilbert space $H{\bigoplus}K\;and\;let\;{\sigma}_w(\cdot)$ denote the Weyl spectrum. We give the necessary and sufficient conditions for operators A and B which ${\sigma}_w\(\array{A & C\\0 & B}\)={\sigma}_w\(\array{A & C\\0 & B}\)\;or\;{\sigma}_w\(\array{A & C\\0 & B}\)={\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.