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# Generalized D-symmetric Operators II

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Published:2010-08-19
Printed: Mar 2011
• S. Bouali,
Department of Mathematics and Informatics, Faculty of Sciences Kénitra, B. P. 133 Kénitra, Morocco
• M. Ech-chad,
Lycée mixte de Missour, 33250 Missour, Morocco
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## Abstract

Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L }(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$. This note will initiate a study on the class of pairs $(A,B)$ such that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal R}(\delta_{A^{\ast}B^{\ast}})}$.
 Keywords: generalized derivation, adjoint, D-symmetric operator, normal operator
 MSC Classifications: 47B47 - Commutators, derivations, elementary operators, etc. 47B10 - Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47A30 - Norms (inequalities, more than one norm, etc.)

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