Let $\mathcal{H}$ be a right quaternionic Hilbert space and let T be a quaternionic normal operator with domain $\mathcal{D}(T)\subset \mathcal{H}$. We prove that there exists a Hilbert basis $\mathcal{N}$ of $\mathcal{H}$, a measure space $({\mathrm{\Omega }}_0,\nu )$, a unitary operator $U:\mathcal{H}\to L^2({\mathrm{\Omega }}_0;\mathbb{H};\nu )$ and a ν-measurable function $\eta :{\mathrm{\Omega }}_0\to \mathbb{C}$ such that$Tx=U^{⁎}{M}_{\eta }Ux,\text{for all}x\in \mathcal{D}(T)$ where $M_{\eta }$ is the multiplication operator on $L^2({\mathrm{\Omega }}_0;\mathbb{H};\nu )$ induced by η with $U(\mathcal{D}(T))\subseteq \mathcal{D}(M_{\eta })$. We show that every complex Hilbert space can be seen as a slice Hilbert space of some quaternionic Hilbert space and establish the main result by reducing the problem to the complex case then lift it to the quaternion case.

让 $\mathcal{H}$是一个正确的四元Hilbert空间，并让T是一个具有域的四元正态算符$\mathcal{d}（Ť）\subset \mathcal{H}$。我们证明存在希尔伯特基础$\mathcal{ñ}$ 的 $\mathcal{H}$，一个测量空间 $（{\mathrm{\Omega }}_0，\nu ）$一元运算符 $ü：\mathcal{H}\to {\mathrm{大号}}^2（{\mathrm{\Omega }}_0;\mathbb{H};\nu ）$和一个ν可测函数$\eta ：{\mathrm{\Omega }}_0\to \mathbb{C}$ 这样$ŤX={ü}^{⁎}{\mathrm{中号}}_{\eta }üX，\text{对所有人}X\in \mathcal{d}（Ť）$ 哪里 ${\mathrm{中号}}_{\eta }$ 是乘法运算符 ${\mathrm{大号}}^2（{\mathrm{\Omega }}_0;\mathbb{H};\nu ）$诱导η与$ü（\mathcal{d}（Ť））\subseteq \mathcal{d}（{\mathrm{中号}}_{\eta }）$。我们证明，每个复杂的希尔伯特空间都可以看作是四元离子希尔伯特空间的一个切片希尔伯特空间，并通过将问题简化为复杂情况，然后将其提升为四元数情况来建立主要结果。