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Operator Theory and Harmonic Analysis

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Twentieth Century Harmonic Analysis — A Celebration

Part of the book series: NATO Science Series ((NAII,volume 33))

Abstract

There are very close ties between the two subjects of the title. In the early days of operator theory, this was mainly manifested in applications of harmonic (and complex) analysis to operator theory. For example, von Neumann proved that the norm of p(T), where T is any contraction on Hubert space andp a polynomial, cannot exceed the maximum of |\p(z) | for z on the unit circle, as an application of complex analysis. In like manner, Stone’s theorems on groups of unitary operators were proved with the aid of Bochner’s theorem characterizing those functions on the circle (or on the reals) which are the Fourier transforms of positive measures. The spectral theorem itself was obtained by various routes based on Bochner’s theorem, the trigonometric and/or algebraic moment theorem, etc.

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Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    H. S. Shapiro

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  1. H. S. Shapiro
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Editors and Affiliations

  1. Prometheus Inc., Newport, RI, USA

    James S. Byrnes

  2. University of Massachusetts at Boston, Boston, MA, USA

    James S. Byrnes

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Shapiro, H.S. (2001). Operator Theory and Harmonic Analysis. In: Byrnes, J.S. (eds) Twentieth Century Harmonic Analysis — A Celebration. NATO Science Series, vol 33. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0662-0_2

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  • DOI: https://doi.org/10.1007/978-94-010-0662-0_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-7169-4

  • Online ISBN: 978-94-010-0662-0

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