On local spectral properties of operator matrices

Abstract

In this paper, we focus on a 2 times 2 operator matrix T_{epsilon _{k}} as follows: \t\t\tTϵk=(ACϵkDB),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} T_{\\epsilon _{k}}= \\begin{pmatrix} A & C \\\\ \\epsilon _{k} D & B\\end{pmatrix}, \\end{aligned}$$ \\end{document} where epsilon _{k} is a positive sequence such that lim_{krightarrow infty }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.