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

 

Essential numerical range of elementary operators


Author: M. Barraa
Journal: Proc. Amer. Math. Soc. 133 (2005), 1723-1726
MSC (2000): Primary 47B47; Secondary 47A12
Published electronically: December 20, 2004
MathSciNet review: 2120257
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Abstract: Let $A= (A_{1},...,A_{p})$ and $B=(B_{1},...,B_{p})$ denote two $p$-tuples of operators with $A_{i}\in \mathcal L(H)$ and $B_{i}\in \mathcal L(K).$ Let $R_{2,A,B}$ denote the elementary operators defined on the Hilbert-Schmidt class $\mathcal C^{2}(H,K)$ by $ R_{2,A,B}(X)=A_{1}XB_{1}+...+A_{p}XB_{p}.$We show that

\begin{displaymath}co\left[(W_{e}(A)\circ W(B))\cup (W(A)\circ W_{e}(B))\right]\subseteq V_{e}(R_{2,A,B}).\end{displaymath}

Here $V_{e}(.)$ is the essential numerical range, $ W(.)$ is the joint numerical range and $W_{e}(.)$ is the joint essential numerical range.


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Additional Information

M. Barraa
Affiliation: Département de Mathématiques, Faculté des Sciences Semlalia, Marrakech, Maroc
Email: barraa@ucam.ac.ma

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07672-5
PII: S 0002-9939(04)07672-5
Keywords: Elementary operators, essential numerical range, Hilbert-Schmidt class
Received by editor(s): November 14, 2003
Received by editor(s) in revised form: February 13, 2004
Published electronically: December 20, 2004
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.