Weyl's and Browder's theorem for an elementary operator

Abstract

Let $\cal H$ be a separable infinite dimensional complex Hilbert space and let$B(\cal H)$ denote the algebra of bounded operators on $\cal H$ into itself. Thegeneralized derivation $\delta_{A,B}$ is defined by $\delta_{A,B}(X)=AX-XB$. For pairs$C=(A_{1},A_{2})$ and $D=(B_{1},B_{2})$ of operators, we define the elementaryoperator $\Phi_{C,D}$ by $\Phi_{C,D}(X)=A_{1}XB_{1}-A_{2}XB_{2}$. If $A_{2}=B_{2}=I$, weget the elementary operator $\Delta_{A_{1},B_{1}}(X)=A_{1}XB_{1}-X$. Let$d_{A,B}=\delta_{A,B}$ or $\Delta_{A,B}$. We prove that if $A, B^{*}$ are$\log$-hyponormal, then $f(d_{A,B})$ satisfies (generalized) Weyl's Theorem for eachanalytic function $f$ on a neighborhood of $\sigma(d_{A,B})$, we also prove that$f(\Phi_{C,D})$ satisfies Browder's Theorem for each analytic function $f$ on aneighborhood of $\sigma(\Phi_{C,D})$.