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 $\delta_{A,B}$.
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