Abstract

A bounded linear operator $T$ acting on a Banach space possesses property (gaw) if $\sigma(T)\setminus E_a(T)=\sigma_{BW}(T)$, where $\sigma_{BW}(T)$ is the B-Weyl spectrum of $T$, $\sigma(T)$ is the usual spectrum of $T$ and $E_a(T)$ is the set of all eigenvalues of $T$ which are isolated in the approximate point spectrum of $T$. In this paper we introduce and study the new spectral properties (z), (gz), (az) and (gaz) as a continuation of [M. Berkani, H. Zariouh, {\it New extended Weyl type theorems}, Mat. Vesnik {\bf 62} (2010), 145--154], which are related to Weyl type theorems. Among other results, we prove that $T$ possesses property (gz) if and only if $T$ possesses property (gaw) and $\sigma_{BW}(T)=\sigma_{SBF_+^-}(T)$; where $\sigma_{SBF_+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$.

Keywords: Property (z); property (gz); property (az); essential semi-B-Fredholm spectrum.

MSC: 47A53, 47A10, 47A11

Pages:  94--103     

Volume  65 ,  Issue  1 ,  2013