The spectral theorem for normal operators on a Clifford module

TL;DR

This paper develops a spectral theorem and Borel functional calculus for normal operators on Clifford modules using the $S$-spectrum, extending spectral theory to non-commuting operator tuples and Dirac operators.

Contribution

It introduces a new spectral theorem and functional calculus for operators on Clifford modules, generalizing existing theories to non-commuting operators and Dirac operators.

Findings

Established spectral theorem for normal operators on Clifford modules.

Developed a Borel functional calculus in the Clifford setting.

Extended spectral theory to non-self adjoint Dirac operators and operator tuples.

Abstract

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.