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The representation of analytic multivalued functions by compact operators

Published online by Cambridge University Press:  24 October 2008

M. C. White
M. C. White
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
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Extract

In this paper we consider the problem of characterizing the variation of the spectrum of a holomorphic family of compact operators ƒ:G → KB(X), where G is an open subset of ℂ and X is a Banach space. The natural conjecture, which the author first heard in a lecture by Professor B. Aupetit, is that these spectra are characterized as those analytic multivalued functions which have as values null sequences. This is obviously a necessary condition, and we prove that this is also sufficient. It will be convenient to use the notation K(ℂ) for the set of compact non-empty subsets of the plane and K0(ℂ) for the subset of K(ℂ) consisting of null sequences.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 161 - 167
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Aupetit, B.. Propriétés Spectrales des Algébres de Banach. Lecture Notes in Math. vol. 735 (Springer-Verlag, 1979).Google Scholar
[2]Aupetit, B.. Analytic multivalued functions in Banach algebras and uniform algebras. Adv. in Math. 44 (1982), 18–60.CrossRefGoogle Scholar
[3]Kaup, L. and Kaup, B.. Holomorphic Functions of Several Variables. Studies in Math. vol. 3 (de Gruyter, 1983).CrossRefGoogle Scholar
[4]Ransford, T. J.. Analytic multivalued functions. Ph.D thesis, Cambridge University (1984).Google Scholar
[5]Rudin, W.. Real and Complex Analysis (Tata McGraw-Hill, 1974).Google Scholar
[6]Slodkowski, Z.. Analytic set-valued functions and spectra. Math. Ann. 256 (1981), 363–383.Google Scholar
[7]Vesentini, E.. On the subharmonicity of the spectral radius. Boll. Un. Mat. Ital. 4 (1968), 427–429.Google Scholar