Remarks on the range and the kernel of generalized derivation

Abstract

Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ and let $\;\mathcal{J}$ denote a two-sided ideal in $L(H)$. Given $A,B\in L(H)$, define the 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 of operators $(A,B)$ 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 $A,B$ for which the pair $(A,B)$ has the property $(PF)_{\mathcal{J}}$. We establish the orthogonality of the range and the kernel of a generalized derivation $\delta_{A,B}$ for non-normal operators $A, B\in L(H)$. We also obtain new results concerning the intersection of the closure of the range and the kernel of $\delta_{A,B}$.

Keywords

• derivation; orthogonality; fuglede-putnam property; subnormal operator; compact operator; dominant operator