New approach to a-Weyl’s theorem and some preservation results

Abstract

In this paper, we introduce and study new spectral properties called (bz) and $(w_{\pi _{00}^{a}})$ in connection with a-Weyl type theorems, which are analogous respectively to a-Weyl’s theorem and a-Browder’s theorem. Among other results, we prove that a bounded linear operator T satisfies property (bz) if and only if T satisfies a-Browder’s theorem and $\sigma _{uf}(T)=\sigma _{uw}(T),$ where $\sigma _{uf}(T)$ and $\sigma _{uw}(T)$ are respectively, the upper semi-Fredholm spectrum and the upper Weyl spectrum. Furthermore, the property (bz) is characterized for an operator T through localized SVEP, and its preservation under direct sum of two bounded linear operators is also examined. The theory is exemplified in the case of some special classes of operators. We also prove the following new result that’s $\sigma _{uf}(T)=\sigma _{uw}(T)\Longleftrightarrow \sigma _{ubf}(T)=\sigma _{ubw}(T),$ for every bounded linear operator T.