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 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.
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