Let L(H) denote the algebra of operators on a complex infinite dimensional Hilbert space H into itself. In this paper, we study the class of operators A ∈ L(H) which satisfy the following property, AT = T A implies AT * = T * A for all T ∈ C1(H) (trace class operators). Such operators are called p-symmetric.We establish some basic properties on the class of p-symmetric operators. We obtain new results concerning the intersection of the closure of R(δA), the range of the derivation δA(X) = AX − XA, and the commutant {A} ′ of A. We introduce the class of essentially d-symmetric operators. Some open problems are also presented.
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A, B ∈ L(H), define the generalized derivation δA, B ∈ L(L(H)) by δA, B(X) = AX - XB. An operator A ∈ L(H) is P-symmetric if AT = TA implies AT* = T* A for all T ∈ C1(H) (trace class operators). In this paper, we give a generalization of P-symmetric operators. We initiate the study of the pairs (A, B) of operators A, B ∈ L(H) such that R(δA, B) W* = R(δA, B) W*, where R(δA, B) W* denotes the ultraweak closure of the range of δA, B. Such pairs of operators are called generalized P-symmetric. We establish a characterization of those pairs of operators. Related properties of P-symmetric operators are also given.
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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