On local spectral properties of operator matrices

Abstract

In this paper, we focus on a 2 x 2 operator matrix T-epsilon k as follows: T-epsilon k = (GRAPHICS), where epsilon(k) is a positive sequence such that lim(k ->infinity) epsilon(k) = 0. We first explore how T-epsilon k has several local spectral properties such as the single-valued extension property, the property (beta), and decomposable. We next study the relationship between some spectra of T-epsilon k and spectra of its diagonal entries, and find some hypotheses by which T-epsilon k satisfies Weyl's theorem and a-Weyl's theorem. Finally, we give some conditions that such an operator matrix T-epsilon k has a nontrivial hyperinvariant subspace.