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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
DOI: https://doi.org/10.1090/S0002-9939-96-03222-4
MathSciNet review: 1307516
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Abstract | References | Similar Articles | Additional Information

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.


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

  • 1. M. D. Acosta and R. Payá, Numerical radius attaining operators and the Radon-Nikodym property, Bull. London Math. Soc. 25 (1993), 67--73. MR 93j:47005
  • 2. I. Berg and B. Sims, Denseness of numerical radius attaining operators, J. Austral. Math. Soc. Ser. A 36 (1984), 130--133. MR 84j:47004
  • 3. F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces, I, II Cambridge Univ. Press, London and New York, 1971, 1973. MR 44:5779
  • 4. D. F. Cudia, The geometry of Banach spaces. Smoothness, Trans. Amer. Math. Soc. 110 (1964), 284--314. MR 29:446
  • 5. G. De Barra, J. R. Giles, and B. Sims, On the numerical range of compact operators on Hilbert spaces, J. London Math. Soc. (2) 5 (1972), 702--706. MR 47:4044
  • 6. N. Dunford and J. T. Schwartz, Linear operators, Part I, General theory, Interscience, New York, 1958. MR 22:8302
  • 7. U. Fixman, F. Okoh, and G. K. R. Rao, An eigenvalue problem for the numerical range of a bounded linear transformation, preprint.
  • 8. P. R. Halmos, Measure theory, Van Nostrand, New York, 1950. MR 11:504d
  • 9. ------, Hilbert space problem book, Van Nostrand, New York, 1967. MR 34:8178
  • 10. L. A. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1971), 1005--1019. MR 46:663
  • 11. G. Lumer, Semi-inner product spaces, Trans. Amer. Math. Soc. 100 (1961), 29--43. MR 24:a2860
  • 12. R. Payá, A counterexample on numerical radius attaining operators, Israel J. Math. 79 (1992), 83--101. MR 93j:47004
  • 13. G. K. R. Rao, Numerical ranges of linear operators in $L_p$-spaces, Doctoral thesis, Queen's University, Kingston, Canada, 1974.
  • 14. V. L. Sm\u{u}lian, Sur la dérivabilité de la norme dans l'espace de Banach, Dokl. Akad. Nauk SSSR 27 (1940), 643--648. MR 2:102f
  • 15. J. P. Williams, Spectra of products and numerical ranges, J. Math. Anal. Appl. 17 (1967), 214--220. MR 34:3341

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

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