Some spectral properties of generalized derivations

Mohamed Amouch, Farida Lombarkia

Given Banach spaces $\cal{X}$ and $\cal{Y}$ and Banach space operators $A\in L(\cal{X})$ and $B\in L(\cal{Y})$, the generalized derivation $\delta_{A,B} \in L(L(\cal{Y},\cal{X}))$ is defined by $\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. This paper is concerned with the problem of 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 an application, we extend some recent results concerning Weyl-type theorems.