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A Survey on the Recent Advances in the Spectral Theory on the S-Spectrum

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Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 290))

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Abstract

The aim of this paper is to give an overview on recent results involving function theory, functional calculi and operator theory based on the S-spectrum. Recently, this prospective has been extended in several directions. In particular, via the S-functional calculus, it is possible to define a theory of slice monogenic functions of a Clifford variable and of noncommuting matrix variables. Moreover, there is a very general way to define the S-functional calculus for Clifford operators and formulate a spectral theorem for Clifford operators based on the S-spectrum.

The first author is partially supported by the PRIN project Direct and inverse problems for partial differential equations: theoretical aspects and applications. Both authors wish to thank the anonymous referee for carefully reading the survey and making a number of suggestions which improved the presentation on the results in the survey.

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Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Fabrizio Colombo

  2. School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, UK

    David P. Kimsey

Authors
  1. Fabrizio Colombo
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  2. David P. Kimsey
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Correspondence to Fabrizio Colombo .

Editor information

Editors and Affiliations

  1. Schmid College of Science & Technology, Chapman University, Orange, CA, USA

    Daniel Alpay

  2. Institut für Angewandte Mathematik, Technische Universität Graz, Graz, Steiermark, Austria

    Jussi Behrndt

  3. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Fabrizio Colombo

  4. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Irene Sabadini

  5. The Donald Bren Distinguished Presidential Chair in Mathematics, Chapman University, Orange, CA, USA

    Daniele C. Struppa

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Colombo, F., Kimsey, D.P. (2023). A Survey on the Recent Advances in the Spectral Theory on the S-Spectrum. In: Alpay, D., Behrndt, J., Colombo, F., Sabadini, I., Struppa, D.C. (eds) Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis. Operator Theory: Advances and Applications, vol 290. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-21460-8_4

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