The Spectral Theorem for Quaternionic Normal Operators

TL;DR

This paper establishes a spectral theorem for bounded normal operators on quaternionic Hilbert spaces, demonstrating the existence and uniqueness of a spectral measure that represents such operators via an integral over their spherical spectrum.

Contribution

It extends the spectral theorem to quaternionic Hilbert spaces, providing a unique spectral measure representation for bounded normal quaternionic operators.

Findings

Existence of a spectral measure for quaternionic normal operators

Representation of operators as integrals over the spherical spectrum

Uniqueness of the spectral measure

Abstract

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a bounded normal right quaternionic linear operator on $\mathcal{H}$. In this paper, we prove that there exists a unique spectral measure $E$ in $\mathcal{H}$ such that $$T=\int_{\sigma_S(T)}\lambda dE_\lambda,$$ where $\sigma_S(T)$ denotes the spherical spectrum of $T$.