Generalized Kato decomposition and essential spectra

Abstract

Let ${\bf R}$ denote any of the following classes: upper (lower) semi-Fredholm operators, Fredholm operators, upper (lower) semi-Weyl operators, Weyl operators, upper (lower) semi-Browder operators, Browder operators. For a bounded linear operator $T$ on a Banach space $X$ we prove that $T=T_M\oplus T_N$ with $T_M \in {\bf R}$ and $T_N$ quasinilpotent (nilpotent) if and only if $T$ admits a generalized Kato decomposition ($T$ is of Kato type) and $0$ is not an interior point of the corresponding spectrum $\sigma_{\bf R}(T)=\{\lambda \in \mathbb{C}: T-\lambda \notin {\bf R}\}$. In addition, we show that every non-isolated boundary point of the spectrum $\sigma_{\bf R}(T)$ belongs to the generalized Kato spectrum of $T$.