Weyl's theorem for a generalized derivation and an elementary operator

Abstract

For $a,bın B(H)$, $B(H)$ the algebra of operators on a complex infinite dimensional Hilbert space $H$, the generalized derivation $\delta_{ab}ın B(B(H))$ and the elementary operator $\triangle_{ab}ın B(B(H))$ are defined by $\delta_{ab}(x)=ax-xb$ and $\triangle_{ab}(x)=axb-x$. Let $d_{ab}=\delta_{ab}$ or $\triangle_{ab}$. It is proved that if $a,b^*$ are hyponormal, then $f(d_{ab})$ satisfies (generalized) Weyl's theorem for each function $f$ analytic on a neighbourhood of $\sigma(d_{ab})$.