Some spectral properties of generalized derivations

Abstract

Given Banach spaces $\Cal{X}$ and $\Cal{Y}$ andBanach space operators $Aın L(\Cal{X})$ and $Bın L(\Cal{Y})$,the generalized derivation $\delta_{A,B} ınL(L(\Cal{Y},\Cal{X}))$ is defined by$\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. This paper is concernedwith the problem of transferring the left polaroid property, fromoperators $A$ and $B^{*}$ to the generalized derivation$\delta_{A,B}$. As a consequence, we give necessary and sufficientconditions for $\delta_{A,B}$ to satisfy generalized a-Browder'stheorem and generalized a-Weyl's theorem. As an application, weextend some recent results concerning Weyl-type theorems.