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.
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