Generalized Kato decomposition and essential spectra

TL;DR

This paper characterizes operators that decompose into a part with specific spectral properties and a quasinilpotent part, linking this to generalized Kato decompositions and the structure of their spectra.

Contribution

It establishes a characterization of operators with generalized Kato decompositions in terms of spectral properties and operator decompositions, extending existing spectral theory.

Findings

Operators decompose into a generalized Kato part and quasinilpotent part iff spectrum conditions are met.

Non-isolated boundary points of the spectrum belong to the generalized Kato spectrum.

Provides spectral criteria for operator decompositions involving various classes of Fredholm and Weyl operators.

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 $T$.