四元数正规算子的谱定理：乘法形式
Spectral Theorem for quaternionic normal operators: Multiplication form
摘要：
设$\mathcal{H}$是右四元数Hilbert空间，$T$是定义域为$\mathcal{D}(T) \subset \mathcal{H}$的四元数正规算子.则对于固定单位虚四元数$m$，存在$\mathcal{H}$的希尔伯特基$\mathcal{N}_{m}$，测度空间$(Ω, μ)$，酉算子$U \colon \mathcal{H} \to L^{2}(Ω; \mathbb{H}; μ)$和$μ$可测函数$ϕ\colon Ω\to \mathbb{C}_m$（这里是$\mathbb{C}_{m} = \{α+ m β; \;α, β\in \mathbb{R}\}$），使得\[ Tx = U^{*}M_ϕUx, \; \mbox{for all}\; x\in \mathcal{D}(T), \]其中$M_ϕ$是$ϕ$与$ U(\mathcal{D}(T)) \subseteq \mathcal{D}(M_ϕ)$在$L^{2}(Ω; \mathbb{H}; μ)$上诱导的乘法算子。在这个过程中，我们证明了每一个复Hilbert空间是一个切片Hilbert空间。我们建立这些结果减少到复杂的情况下，然后解除它的四元数的情况下。
英文摘要：
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 $(Ω, μ)$, a unitary operator $U \colon \mathcal{H} \to L^{2}(Ω; \mathbb{H}; μ)$ and a $μ$ - measurable function $ϕ\colon Ω\to \mathbb{C}_m$ (here $\mathbb{C}_{m} = \{α+ m β; \;α, β\in \mathbb{R}\}$) such that \[ Tx = U^{*}M_ϕUx, \; \mbox{for all}\; x\in \mathcal{D}(T), \] where $M_ϕ$ is the multiplication operator on $L^{2}(Ω; \mathbb{H}; μ)$ induced by $ϕ$ with $ U(\mathcal{D}(T)) \subseteq \mathcal{D}(M_ϕ)$. 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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