The Spectral Theorem for Quaternionic Normal Operators

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 $T$.