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


Domaine numérique du produit et de la bimultiplication $M_{2,A,B}$

Author: Mohamed Chraibi Kaadoud
Journal: Proc. Amer. Math. Soc. 132 (2004), 2421-2428
MSC (2000): Primary 47A12
Published electronically: March 3, 2004
MathSciNet review: 2052420
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Abstract: In this paper, we present an extension of Bouldin's result (1970) concerning the numerical range $W(AB)$ of the product of two operators $A$ and $B$ that are commuting and for which one of the set $W(A)$ or $ W(B) $ consists of positive numbers. We also prove that if $A$ or $B$ is a subnormal operator on a separable Hilbert space, then

\begin{displaymath}\overline{W(M_{2,A,B})}=\overline{co\left[ W(A)W(B)\right] }, \end{displaymath}

where $M_{2,A,B}$ is the operator bimultiplication and $co$ is the convex hull.

RÉSUMÉ. Dans ce travail, nous améliorons un résultat de Bouldin (1970) concernant la localisation de $W(AB),$le domaine numérique du produit de deux opérateurs $A$ et $B$ sur un espace de Hilbert lorsque $A$ et $B$ commutent et $W(A)$ est constitué de réels strictement positifs. Dans le cas où $A$ ou $B$ est un opérateur sous normal sur un espace de Hilbert séparable, nous montrons que

\begin{displaymath}\overline{W(M_{2,A,B})}=\overline{co\left[ W(A)W(B)\right] }, \end{displaymath}

$M_{2,A,B}$ est l'opérateur produit ou bimultiplication et $co$est l'enveloppe convexe.

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Additional Information

Mohamed Chraibi Kaadoud
Affiliation: Département des Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakech, Maroc

PII: S 0002-9939(04)07352-6
Keywords: Domaine num\'{e}rique, bimultiplication
Received by editor(s): June 19, 2002
Received by editor(s) in revised form: May 16, 2003
Published electronically: March 3, 2004
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society