Limit Points for Browder Spectrum of Operator Matrices

Abstract

Let $A\in \mathcal{B}(X)$ and $B\in \mathcal{B}(Y)$, where $X$ and $Y$ are Banach spaces, and let $M_C$ be an operator acting on $X\oplus Y$ given by $M_C=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right)$. We investigate the limit point set of the Browder spectrum of $M_C$. It is shown that $$acc{\sigma }_b(M_C)\cup W_{acc{\sigma }_b}=acc{\sigma }_b(A)\cup acc{\sigma }_b(B)$$ where $W_{acc{\sigma }_b}$ is a subsets of $acc{\sigma }_{\ast }(B)\cap acc{\sigma }_{\ast }(A)$ and a union of certain holes in $acc{\sigma }_b(M_C)$. Furthermore, several sufficient conditions for $acc{\sigma }_b(M_C)=acc{\sigma }_b(A)\cup acc{\sigma }_b(B)$ holds for every $C\in \mathcal{B}(Y,X)$ are given.