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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
DOI: https://doi.org/10.1090/S0002-9939-04-07352-6
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.


References [Enhancements On Off] (What's this?)

  • 1. T. Ando, Distance to the set of thin operators, preprint, 1970.
  • 2. E. Bishop, Spectral theory for operators on a Banach space, Trans. Amer. Math. Soc., 86 (1957), 414-445. MR 20:7217
  • 3. R. Bouldin, The numerical range of a product, J. Math. Anal. Appl. 32 (1970), 459-467. MR 42:5079a
  • 4. R. Bouldin, The numerical range of a product, II, J. Math. Anal. Applic. 33 (1971), 212-219. MR 42:5079a
  • 5. M. K. Chraibi, Domaine numérique de l'opérateur produit $M_{2,A,B}$ et de la dérivation généralisée $\delta _{2,A,B}$, Extracta Mathematicae. Vol. 17, Num. 1, 2002, 59-68. MR 2003h:47065
  • 6. M. K. Chraibi, Domaine numérique d'op érateurs élémentaires. Problèmes de rafle, thèse d' état, Université Cadi Ayyad, Faculté des Sciences Semlalia. Marrakech. 2002. N$^{\circ }$ D'ordre 339.
  • 7. L. Fialkow, Structural properties of elementary operators, Elementary operators and applications (Blaubeuren, 1991), 55-113, World Sci. Publishing, River Edge, NJ, 1992. MR 93i:47042
  • 8. C. K. Fong, Most normal operators are diagonal, Proc. Amer. Math. Soc. 99 (1987), 671-672. MR 88b:47033
  • 9. K. Gustafson and D. Rao, The field of values of linear operators and matrices, Universitext, Springer-Verlag, New York, 1997.MR 98b:47008
  • 10. P. R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, NJ, 1967. MR 34:8178
  • 11. J. A. R. Holbrook, On the power bounded operators of Sz.-Nagy and Foias, Acta Sci. Math. (Szeged) 29 (1968), 299-310. MR 39:810
  • 12. C. K. Li, C-numerical range and C-numerical radii, Linear and Multilinear Algebra 37 (1994), 51-82. MR 95k:15039
  • 13. G. Lumer and M. Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32-41. MR 21:2927
  • 14. R. Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, 27, 2nd edition, Springer-Verlag, Berlin-Heidelberg-New York, 1970. MR 41:2449

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Additional Information

Mohamed Chraibi Kaadoud
Affiliation: Département des Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakech, Maroc
Email: chraibik@ucam.ac.ma

DOI: https://doi.org/10.1090/S0002-9939-04-07352-6
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

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