Generalized Kato-Riesz decomposition

TL;DR

This paper introduces the concept of generalized Kato-Riesz decomposition for bounded linear operators, explores the associated spectrum, and characterizes operators with this property, including generalized Drazin-Riesz invertibility.

Contribution

It defines and investigates the generalized Kato-Riesz spectrum and characterizes operators as direct sums of Riesz and semi-Fredholm operators, extending spectral theory.

Findings

Characterization of generalized Drazin-Riesz invertible operators

Description of operators as direct sums involving Riesz and semi-Fredholm operators

Analysis of the single-valued extension property in this context

Abstract

We shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-Riesz decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and $T_N$ is Riesz. In this paper we define and investigate the generalized Kato-Riesz spectrum of an operator. For $T$ is said to be generalized Drazin-Riesz invertible if there exists a bounded linear operator $S $ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is Riesz. We investigate generalized Drazin-Riesz invertible operators and also, characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular we characterize the single-valued extension property at a point $\lambda_0\in{\mathbb C}$ in the case that $\lambda_0-T$ admits a generalized Kato-Riesz decomposition.