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

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).