The spectral theorem for normal operators on a Clifford module

Fabrizio Colombo; David P. Kimsey

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arXiv · 2112.05201

The spectral theorem for normal operators on a Clifford module

Fabrizio Colombo·David P. Kimsey

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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.

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Fabrizio Colombo, David P. Kimsey. 2021-12-09. The spectral theorem for normal operators on a Clifford module. https://arxiv.org/abs/2112.05201

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