Abstract

BOUHAFSI, Youssef; ECH-CHAD, Mohamed  and  ZOUAKI, Adil. A Note on the Range of a Derivation. Rev.colomb.mat. [online]. 2022, vol.56, n.2, pp.145-155.  Epub Jan 03, 2024. ISSN 0034-7426.  https://doi.org/10.15446/recolma.v56n2.108371.

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 δ A, B ∈ L(L(H)) by δ A, B (X) = AX - XB. An operator A ∈ L(H) is P-symmetric if AT = TA implies AT * = T * A for all T ∈ C 1(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(δ A, B ) W* = R(δ A, B ) W* , where R(δ A, B ) W* denotes the ultraweak closure of the range of δ A, 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.

Keywords : Generalized derivation; Fuglede-Putnam property; D-symmetric operator; P-symmetric operator; Compact operator.

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