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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.56 no.2 Bogotá July/Dec. 2022 Epub Jan 03, 2024
https://doi.org/10.15446/recolma.v56n2.108371
Original articles
A Note on the Range of a Derivation
Una nota sobre el rango de una derivada
1 Université Chouaib Doukkali, El Jadida, Marruecos
2 Université Ibn Tofail, Kenitra, Marruecos
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
Sea H un espacio de Hilbert separable sobre los complejos y denote por L(H) al álgebra de los operadores acotados de H es sí mismo. Dados A, B ∈ L(H), defina la derivada generalizada δ A, B ∈ L(L(H)) como δ A, B (X) = AX - XB. Un operador A ∈ L(H) es P-simétrico si la condición AT = TA implica que AT* = T* A para todo T ∈ C 1(H) (los operadores de clase de traza). En este artículo presentamos una generalización de los operadores P-simétricos. En este artículo estudiamos pares (A, B) de operadores A, B ∈ L(H) tales que R(δ A, B ) W* = R(δ A, B ) W* , donde R(δ A, B ) W* denota la clausura ultradébil del rango δ A, B . A esta clase de operadores los llamamos operadores P-simétricos generalizados. En este artículo damos una caracterización de esta clase de pares de operadores y presentamos propiedades de los operadores P-simétricos generalizados.
Palabras clave: Derivada generalizada; propiedad de Fuglede-Putnam; operador D-simétrico; operador P-simétrico; operador compacto
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Received: January 31, 2022; Accepted: October 17, 2022
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