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On the range of a generalized derivation

Authors:	S Mecheri

Institution:	(1) School of Mathematics, The University of New South Wales, Sydney, 2052, Australia;(2) Department of Mathematics, 333 Avery Hall, The University of Nebraska, Lincoln, Lincoln, NE 68588-0130, USA;(3) Department of Mathematics, Creighton University, Omaha, NE 68178, USA

Abstract:	A generalized derivation $\delta _{{\rm A},{\rm B}} :\mathcal{L}(\mathcal{H}) \mapsto \mathcal{L}(\mathcal{H})$ , is defined by the formula $\delta _{{\rm A},{\rm B}} :(X) = AX - XB$ , where ${\rm A},{\rm B} \in \mathcal{L}(\mathcal{H})$ and $\mathcal{L}(\mathcal{H})$ is the Banach algebra of bounded linear operators in a Hilbert space $\mathcal{H}$ . Sufficient conditions under which $\overline {R(\delta _{{\rm A},{\rm B}} )} \cap ker\delta _{{\rm A},{\rm B}} = \left\{ 0 \right\}$ and $\overline {R(\delta _{{\rm A},{\rm B}} )} \cap ker\delta _{{\rm A}^ * ,{\rm B}^ * } = \left\{ 0 \right\}$ are given. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 111–119.

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