Some spectral properties for generalized derivations

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.