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Semi-normal operators on uniformly smooth Banach spaces

  • Muneo Chō (a1)

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In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denote

D(x) = {f ε X*:∥f∥ = f(x) = l}.

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References

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1.de Barra, G., Some algebras of operators with closed convex numerical range, Proc. Roy. Irish Acad. 72 (1972), 149154.
2.de Barra, G., Generalized limits and uniform convexity. Proc. Roy. Irish Acad. 74 (1974), 7377.
3.Beauzamy, B., Introduction to Banach spaces and their geometry (North-Holland, 1985).
4.Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and of elements of normed algebras (Cambridge, 1971).
5.Bonsall, F. F. and Duncan, J., Numerical ranges II (Cambridge, 1973).
6.Chō, M., Joint spectra of operators on Banach space, Glasgow Math. J. 28 (1986), 6972.
7.Chō, M., Joint spectra of commuting normal operators on Banach spaces, Glasgow Math J. 30 (1988), 339345.
8.Chō, M., Hyponormal operators on uniformly convex spaces, Acta Sci. Math. (Szeged), to appear.
9.Chō, M. and Dash, A. T., On the joint spectra of doubly commuting n-tuples of semi-normal operators, Glasgow Math. J. 26 (1985), 4750.
10.Chō, M. and Yamaguchi, H., Bare points of joint numerical ranges for doubly commuting hyponormal operators on strictly c-convex spaces, preprint.
11.Mattila, K., Normal operators and proper boundary points of the spectra of operators on Banach space, Ann. Acad. Sci. Fenn. AI Math. Dissertationes 19 (1978).
12.Mattila, K., Complex strict and uniform convexity and hyponormal operators, Math. Proc. Cambridge Philos. Soc. 96 (1984), 483497.
13.Putnam, C. R., Commutation properties of Hilbert space operators and related topics. (Springer, 1967).
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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