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Cauchy’s Singular Integral Operator and Its Beautiful Spectrum

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Systems, Approximation, Singular Integral Operators, and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 129))

Abstract

The Cauchy singular integral operator, S, is one of the main actors in the theory of Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations. While the boundedness of S has been studied for many decades, final results on the spectrum of S were obtained only in recent times. During the last few years, it was discovered that there is a surprising and undreamt-of metamorphosis of the (local) spectra of S from circular arcs via horns and logarithmic double-spirals to so-called logarithmic leaves with a halo. This article is a survey of this fascinating development. We hope it is both a feast for the eyes and an illustration of Nick Trefethen’s motto: The spectrum gives an operator a personality!

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Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    A. Böttcher

  2. Departamento de Matemáticas, CINVESTAV del I.P.N., Apartado Postal 14-740, 07000, México, D.F., México

    Yu. I. Karlovich

Authors
  1. A. Böttcher
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  2. Yu. I. Karlovich
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Editors and Affiliations

  1. Département de Mathématiques, Université de Bordeaux I, 351, cours de la Libération, 33405, Talence, France

    Alexander A. Borichev  & Nikolai K. Nikolski  & 

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Böttcher, A., Karlovich, Y.I. (2001). Cauchy’s Singular Integral Operator and Its Beautiful Spectrum. In: Borichev, A.A., Nikolski, N.K. (eds) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8362-7_5

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  • DOI: https://doi.org/10.1007/978-3-0348-8362-7_5

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9534-7

  • Online ISBN: 978-3-0348-8362-7

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