The spectral theorem for normal operators on a Clifford module
Abstract
In this paper, using the recently discovered notion of the -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 ). 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 -tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of -tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.
Cite
@article{arxiv.2112.05201,
title = {The spectral theorem for normal operators on a Clifford module},
author = {Fabrizio Colombo and David P. Kimsey},
journal= {arXiv preprint arXiv:2112.05201},
year = {2021}
}