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

Abstract

WhenA∈B(H)andB∈B(K)are given, we denote byMCthe operator acting on the infinite-dimensional separable Hilbert spaceH⊕Kof the formMC=(AC0B). In this paper, it is proved that there exists some operatorC∈B(K,H)such thatMCis upper semi-Browder if and only if there exists some left invertible operatorC∈B(K,H)such thatMCis upper semi-Browder. Moreover, a necessary and sufficient condition forMCto be upper semi-Browder for someC∈G(K,H)is given, whereG(K,H)denotes the subset of all of the invertible operators ofB(K,H).