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

Abstract

Let $\mathcal{H}$ be a separable infinite dimensional complex Hilbert space and let$B(\mathcal{H}\mathcal{)}$ denote the algebra of bounded operators on $\mathcal{H}$ 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 ${\mathrm{\Phi }}_{C,D}$ by ${\mathrm{\Phi }}_{C,D}(X)=A_{1}X{B}_1-A_{2}X{B}_2$. If $A_2=B_2=I$, weget the elementary operator ${\mathrm{\Delta }}_{A_1,B_1}(X)=A_{1}X{B}_1-X$. Let$d_{A,B}={\delta }_{A,B}$ or ${\mathrm{\Delta }}_{A,B}$. We prove that if $A,B^{\ast }$ 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$f({\mathrm{\Phi }}_{C,D})$ satisfies Browder's Theorem for each analytic function $f$ on aneighborhood of $\sigma ({\mathrm{\Phi }}_{C,D})$.