Some spectral properties of generalized derivations

Abstract

Given Banach spaces $\mathcal{X}$ and $\mathcal{Y}$ andBanach space operators $AınL(\mathcal{X}\mathcal{)}$ and $BınL(\mathcal{Y}\mathcal{)}$,the generalized derivation ${\delta }_{A,B}ınL(L(\mathcal{Y}\mathcal{,}\mathcal{X}\mathcal{)}\mathcal{)}$ 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 $B^{\ast }$ 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.