Schur Analysis in the Quaternionic Setting: The Fueter Regular and the Slice Regular Case

Daniel Alpay (), Fabrizio Colombo () and Irene Sabadini ()
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Daniel Alpay: Ben-Gurion University of the Negev, Earl Katz Chair in Algebraic System Theory, Department of Mathematics
Fabrizio Colombo: Politecnico di Milano, Dipartimento di Matematica
Irene Sabadini: Politecnico di Milano, Dipartimento di Matematica

Chapter 62 in Operator Theory, 2015, pp 1745-1786 from Springer

Abstract: Abstract This chapter is a survey on recent developments in quaternionic Schur analysis. The first part is based on functions which are slice hyperholomorphic in the unit ball of the quaternions, and have modulus bounded by 1. These functions, which by analogy to the complex case are called Schur multipliers, are shown to be (as in the complex case) the source of a wide range of problems of general interest. They also suggest new problems in quaternionic operator theory, especially in the setting of indefinite inner product spaces. This chapter gives an overview on rational functions and their realizations, on the Hardy space of the unit ball, on the half-space of quaternions with positive real part, and on Schur multipliers, also discussing related results. For the purpose of comparison this chapter presents also another approach to Schur analysis in the quaternionic setting, in the framework of Fueter series. To ease the presentation most of the chapter is written for the scalar case, but the reader should be aware that the appropriate setting is often that of vector-valued functions.

Keywords: Schur Analysis; Quaternionic Setting; Schur Multiplier; Hardy Space; Slice Hyperholomorphic Functions (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0667-1_20

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DOI: 10.1007/978-3-0348-0667-1_20

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