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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the range and the kernel
of the operator $X\mapsto AXB-X$


Author: A. Mazouz
Journal: Proc. Amer. Math. Soc. 127 (1999), 2105-2107
MSC (1991): Primary 47B47, 47D50
Published electronically: March 3, 1999
MathSciNet review: 1487327
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Abstract: Let $L(H)$ denote the algebra of (bounded linear) operators on the separable complex Hilbert space $H$, and let $(\mathfrak I;\|\,.\,\|_{\mathfrak I})$ denote a norm ideal in $L(H)$. For $A,B\in L(H)$, let the derivation $\delta _{A,B}\colon L(H)\to L(H)$ be defined by $\delta _{A,B}(X)=AX-XB$, and let $\Delta _{A,B}:L(H)\to L(H)$ be defined by $\Delta _{A,B}(X)=AXB-X$. The main result of this paper is to show that if $A$, $B$ are contractions, then for every operator $T\in\mathfrak J$ such that $ATB=T$, then $\|AXB-X+T\|_{\mathfrak J}\ge \|T\|_{\mathfrak J}$ for all $X\in\mathfrak J$.


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Additional Information

A. Mazouz
Affiliation: Département de Mathématiques, Université Montpellier II, Place E.-Bataillon, 34060 Montpellier Cedex, France

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04754-1
PII: S 0002-9939(99)04754-1
Received by editor(s): December 2, 1996
Received by editor(s) in revised form: October 16, 1997
Published electronically: March 3, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society