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Universality property of the S-functional calculus, noncommuting matrix variables and Clifford operators

Lookup NU author(s): Dr David KimseyORCiD

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).

Abstract

© 2022 Elsevier Inc.Spectral theory on the S-spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S-spectrum has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n-tuples of noncommuting operators. With this paper we show that the spectral theory on the S-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. In fact, the S-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S-functional calculus goes beyond quaternionic and Clifford analysis, indeed the S-functional calculus has a certain universality property. This fact makes the spectral theory on the S-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

Publication metadata

Author(s): Colombo F, Gantner J, Kimsey DP, Sabadini I

Publication type: Article

Publication status: Published

Journal: Advances in Mathematics

Year: 2022

Volume: 410

Issue: Part A

Print publication date: 03/12/2022

Online publication date: 13/10/2022

Acceptance date: 28/09/2022

Date deposited: 19/12/2022

ISSN (print): 0001-8708

ISSN (electronic): 1090-2082

Publisher: Academic Press Inc.

URL: https://doi.org/10.1016/j.aim.2022.108719

DOI: 10.1016/j.aim.2022.108719

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