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The spectral theorem for normal operators on a Clifford module

Lookup NU author(s): Dr David KimseyORCiD

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Abstract

© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be - 1). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

Publication metadata

Author(s): Colombo F, Kimsey DP

Publication type: Article

Publication status: Published

Journal: Analysis and Mathematical Physics

Year: 2022

Volume: 12

Issue: 1

Print publication date: 01/02/2022

Online publication date: 27/12/2021

Acceptance date: 18/11/2021

ISSN (print): 1664-2368

ISSN (electronic): 1664-235X

Publisher: Birkhauser

URL: https://doi.org/10.1007/s13324-021-00628-8

DOI: 10.1007/s13324-021-00628-8

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