Generalized Kato decomposition and essential spectra

Abstract

Let $\mathbf{R}$ 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 $T_M\in \mathbf{R}$ 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 ${\sigma }_{\mathbf{R}}(T)=\{\lambda \in \mathbb{C}:T-\lambda \notin \mathbf{R}\}$. In addition, we show that every non-isolated boundary point of the spectrum ${\sigma }_{\mathbf{R}}(T)$ belongs to the generalized Kato spectrum of $T$.

25 pages