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Generalized D-symmetric Operators II |
Canad. Math. Bull. 54(2011), 21-27
Published:2010-08-19
Printed: Mar 2011
Printed: Mar 2011
- S. Bouali,
- M. Ech-chad,
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.) |