Let L(H) denote the algebra of operators on a complex infinite dimensional Hilbert space H into itself. For A, B \in L(H), the elementary operator \tau A,B \in L(L(H)) is defined by \tau A,B (X) = AXB -X. An operator A \in L(H) is said to be generalized quasi-adjoint if AT A = T implies A \ast T A \ast = T for every T \in C 1 (H) (trace class operators). In this paper, we give an extension of generalized quasi-adjoint operators. We consider the class of pairs of operators A, B \in L(H), where R(\tau A,B )W \ast denotes the ultra-weak closure of the range R(\tau A,B ) of \tau A,B . Such pairs of operators are called generalized quasi-adjoint. We establish some basic properties of those pairs of operators.Нехай L(H) -алгебра операторiв у комплексному нескiнченновимiрному гiльбертовому просторi H. Для A, B \in L(H), елементарний оператор \tau A,B \inпозначене ультраслабке замикання областi значень R(\tau A,B ) of \tau A,B . Такi пари операторiв звуться узагальненими квазiспряженими. Встановленi основнi властивостi таких пар операторiв.
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 $\delta_{A,B}$.
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