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Analytic multifunctions and their applications

  • Chapter
Complex Potential Theory

Part of the book series: NATO ASI Series ((ASIC,volume 439))

Abstract

This long survey article is a quick introduction to the theory of analytic multifunctions and their numerous applications. This new theory originates both from spectral theory and several complex variables. In the introduction we describe its historical evolution from its very beginnings to the present day. Then are given the main results of the general theory of analytic multifunctions like Liouville’s theorem, the localization principle, the holomorphic variation of isolated points, the scarcity theorems for finite or countable analytic multifunctions, Slodkowski’s theorem, the Oka-Nishino theorem and finally the open mapping theorem and Picard’s theorem on distribution of values. In the rest of the text are first given very striking applications to spectral theory, then to joint spectrum, to uniform algebras in connection with problems of analytic structure and polynomial and entire approximation in ℂn, to spectral interpolation and to local spectrum. Recently, important applications were given to non-associative Jordan-Banach algebras and to complex dynamics, that is, the study of the variation of Julia sets depending on a parameter, which are described in the last two chapters.

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Authors and Affiliations

  1. Département de mathématiques et de statistique, Université Laval, Québec, G1K 7P4, Canada

    Bernard Aupetit

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  1. Bernard Aupetit
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  1. Département de Mathématiques et de Statistique, and Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec, Canada

    Paul M. Gauthier

  2. Département de Mathématiques et de Statistique, Faculté des Arts et des Sciences, Université de Montréal, Montréal, Québec, Canada

    Gert Sabidussi

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© 1994 Springer Science+Business Media Dordrecht

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Aupetit, B. (1994). Analytic multifunctions and their applications. In: Gauthier, P.M., Sabidussi, G. (eds) Complex Potential Theory. NATO ASI Series, vol 439. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0934-5_1

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  • DOI: https://doi.org/10.1007/978-94-011-0934-5_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4403-5

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