Generalized Browder's and Weyl's Theorems for Generalized Derivations

TL;DR

This paper extends Browder's and Weyl's theorems to generalized derivations on Banach space operators, characterizing spectra and isolated points, and establishing properties like polaroid and isoloid under certain conditions.

Contribution

It provides a comprehensive analysis of spectral properties and theorems for generalized derivations, including new characterizations and property transfers.

Findings

Characterization of isolated points of the spectrum of generalized derivations

Extension of polaroid and isoloid properties to generalized derivations

Establishment of Weyl and Browder type theorems for these operators

Abstract

Given Banach spaces $\X$ and $\Y$ and Banach space operators $A\in L(\X)$ and $B\in L(\Y)$, let $\rho\colon L(\Y,\X)\to L(\Y,\X)$ denote the generalized derivation defined by $A$ and $B$, i.e., $\rho (U)=AU-UB$ ($U\in L(\Y,\X)$). The main objective of this article is to study Weyl and Browder type theorems for $\rho\in L(L(\Y,\X))$. To this end, however, first the isolated points of the spectrum and the Drazin spectrum of $\rho\in L(L(\Y,\X))$ need to be characterized. In addition, it will be also proved that if $A$ and $B$ are polaroid (respectively isoloid), then $\rho$ is polaroid (respectively isoloid).