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

Abstract EN

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite-dimensional separable Hilbert space H⊕K of the form MC=(AC0B).

In this paper, it is proved that there exists some operator C∈B(K,H) such that MC is upper semi-Browder if and only if there exists some left invertible operator C∈B(K,H) such that MC is upper semi-Browder.

Moreover, a necessary and sufficient condition for MC to be upper semi-Browder for some C∈G(K,H) is given, where G(K,H) denotes the subset of all of the invertible operators of B(K,H).