Some spectral properties for generalized derivations

TL;DR

This paper investigates spectral properties of generalized derivations on Banach spaces, focusing on the transfer of the left polaroid property and conditions for generalized Browder's and Weyl's theorems.

Contribution

It establishes conditions under which spectral properties transfer from operators to their associated generalized derivations, extending Weyl type theorems.

Findings

Transfer of left polaroid property under certain conditions

Necessary and sufficient conditions for generalized a-Browder's theorem

Extension of recent Weyl type theorems

Abstract

Given Banach spaces $\mathcal{X}$ and $\mathcal{Y}$ and Banach space operators $A\in L(\mathcal{X})$ and $B\in L(\mathcal{Y}).$ The generalized derivation $\delta_{A,B} \in L(L(\mathcal{Y},\mathcal{X}))$ is defined by $\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. This paper is concerned with the problem of the transferring the left polaroid property, from operators $A$ and $B^{*}$ to the generalized derivation $\delta_{A,B}$. As a consequence, we give necessary and sufficient conditions for $\delta_{A,B}$ to satisfy generalized a-Browder's theorem and generalized a-Weyl's theorem. As application, we extend some recent results concerning Weyl type theorems.