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 $U:\mathcal{H}\to L^2(\Omega ;\mathbb{H};\mu )$ and a $\mu$ - measurable function $\varphi :\Omega \to {\mathbb{C}}_m$ (here ${\mathbb{C}}_m=\{\alpha +m\beta ;\alpha ,\beta \in \mathbb{R}\}$) such that \[ Tx = U^{*}M_ϕUx, \; \mbox{for all}\; x\in \mathcal{D}(T), \] where $M_{\varphi }$ is the multiplication operator on $L^2(\Omega ;\mathbb{H};\mu )$ induced by $\varphi$ with $U(\mathcal{D}(T))\subseteq \mathcal{D}(M_{\varphi })$. 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.

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References in corpus (1)

• Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures

Cited by in corpus (1)

• Continuous Functional Calculus for Quaternionic Bounded Normal Operators