Remarks on the range and the kernel of generalized derivation

Abstract

Let $L(H)$ denote the algebra of operators on a complexinfinite dimensional Hilbert space $H$ and let $\;\mathcal{J}$denote a two-sided ideal in $L(H)$. Given $A,B\in L(H)$, definethe generalized derivation $\delta_{A,B}$ as an operator on$L(H)$ by
 \centerline{$\delta_{A,B}(X)=AX-XB.$}
 \smallskip\noi We say that the pair ofoperators $(A,B)$ has the Fuglede-Putnam property$(PF)_{\mathcal{J}}$ if $AT=TB$ and $T\in \mathcal{J}$ implies$A^{\ast}T=TB^{\ast}$. In this paper, we give operators $A,B$ forwhich the pair $(A,B)$ has the property $(PF)_{\mathcal{J}}$. Weestablish the orthogonality of the range and the kernel of ageneralized derivation $\delta_{A,B}$ for non-normal operators $A,B\in L(H)$. We also obtain new results concerning the intersectionof the closure of the range and the kernel of $\delta_{A,B}$.