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Joint spectra of operators on Banach space

  • Muneo Chō (a1)

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Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that

(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

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References

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1.Bonsall, F. F. and Duncan, J., Numerical ranges of operators on normed spaces and elements of normed algebras (Cambridge, 1971).
2.Bonsall, F. F. and Duncan, J., Numerical ranges II (Cambridge, 1973).
3.Chō, M. and Takaguchi, M., Some classes of commuting n-tuple of operators, Studia Math., to appear.
4.Clarkson, J. A., Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396414.
5.Crabb, M. J., The numerical range of an operator, Ph.D. thesis, University of Edinburgh, 1969.
6.Dekker, N. P., Joint numerical range and joint spectrum of Hilbert space operators, Ph.D. thesis, University of Amsterdam, 1969.
7.Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 2943.
8.Taylor, J. L., A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172191.
9.Williams, J. P., Spectra of product and numerical ranges, J. Math. Anal. Appl. 17 (1967), 214220.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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