The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

Abstract

When $A\in B(H)$ and $B\in B(K)$ are given, we denote by $M_C$ the operator acting on the infinite-dimensional separable Hilbert space $H\oplus K$ of the form . In this paper, it is proved that there exists some operator $C\in B(K,H)$ such that $M_C$ is upper semi-Browder if and only if there exists some left invertible operator $C\in B(K,H)$ such that $M_C$ is upper semi-Browder. Moreover, a necessary and sufficient condition for $M_C$ to be upper semi-Browder for some $C\in G(K,H)$ is given, where $G(K,H)$ denotes the subset of all of the invertible operators of $B(K,H)$.