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

 

The existence of a maximizing vector
for the numerical range of a compact operator


Authors: Uri Fixman, Frank Okoh and G. K. R. Rao
Journal: Proc. Amer. Math. Soc. 124 (1996), 1133-1138
MSC (1991): Primary 47A12
MathSciNet review: 1307516
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Abstract: Let $X$ be a complex Lebesgue space with a unique duality map $J$ from $X$ to $X^*$, the conjugate space of $X$. Let $A$ be a compact operator on $X$. This paper focuses on properties of $W(A)=\{J(x)(A(x)):\|x\|=1\}$ and $ \Lambda(A)=\sup\{\RE\alpha:\alpha\in W(A)\}$. We adapt an example due to Halmos to show that for $X=l_p, 1<p<\infty$, there is a compact operator $A$ on $l_p$ with $W(A)$ the semi-open interval $[-1,0)$. So $\Lambda(A)$ is not attained as a maximum. A corollary of the main result in this paper is that if $X=l_p,1<p<\infty$, and $\Lambda(A)\ne 0$, then $\Lambda(A)$ is attained as a maximum.


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Additional Information

Uri Fixman
Affiliation: (U. Fixman and G. K. R. Rao) Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6

Frank Okoh
Email: okoh@math.wayne.edu

G. K. R. Rao
Affiliation: (F. Okoh) Department of Mathematics, Wayne State University, Detroit, Michigan 48202

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03222-4
PII: S 0002-9939(96)03222-4
Received by editor(s): October 3, 1994
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society