Spectral Theorem for quaternionic normal operators: Multiplication form

Abstract

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis $\mathcal{N}_{m}$ of $\mathcal{H}$, a measure space $(\Omega, \mu)$, a unitary operator $U \colon \mathcal{H} \to L^{2}(\Omega; \mathbb{H}; \mu)$ and a $\mu$ - measurable function $\phi \colon \Omega \to \mathbb{C}_m$ (here $\mathbb{C}_{m} = \{\alpha + m \beta; \;\alpha, \beta \in \mathbb{R}\}$) such that \[ Tx = U^{*}M_{\phi}Ux, \; \mbox{for all}\; x\in \mathcal{D}(T), \] where $M_{\phi}$ is the multiplication operator on $L^{2}(\Omega; \mathbb{H}; \mu)$ induced by $\phi$ with $ U(\mathcal{D}(T)) \subseteq \mathcal{D}(M_{\phi})$. In the process, we prove that every complex Hilbert space is a slice Hilbert space. We establish these results by reducing it to the complex case then lift it to the quaternionic case.