The spectral theorem for a normal operator on a Clifford module

In this talk we will consider the problem of obtaining a spectral resolution for a densely defined closed normal operator on a Clifford module $\mathcal{H}_n := \mathcal{H} \otimes \mathbb{R}_n$, where $\mathcal{H}$ is a real Hilbert space and $\mathbb{R}_n := \mathbb{R}_{0, n}$ is the Clifford algebra generated by the units $e_1, \ldots, e_n$ with $e_i e_j = -e_j e_i$ for $i \neq j$ and $e_j^2 = -1$ for $j=1,\ldots, n$. We shall see that any densely defined closed normal operator on a Clifford module admits an integral representation which is analogous to the integral representation for a densely defined closed normal operator on a quaternionic Hilbert space (which one may think of as a Clifford module $\mathcal{H}_2$) discovered by Daniel Alpay, Fabrizio Colombo and the speaker in 2014. However, the Clifford module setting sketched above with $n > 2$ presents a number of technical difficulties which are not present in the quaternionic Hilbert space case.

In order to prove this result, one needs to slightly generalise the notion of $S$-spectrum to allow for operators which are not necessarily paravector operators, i.e., operators of the form $T =T_0 + \sum_{j=1}^n T_j e_j$. This observation has implications on a generalisation of the $S$-functional calculus and some related function theory which we shall briefly highlight.

The main thrust of this talk is based on joint work with Fabrizio Colombo. The work on the $S$-functional calculus is joint work with Fabrizio Colombo, Jonathan Gantner and Irene Sabadini. The work on the related function theory is joint work with Fabrizio Colombo, Irene Sabadini and Stefano Pinton.