Let $H$ be a nonzero complex [Hilbert space](/page/Hilbert%20Space), let $T \in \mathcal{L}(H)$ be normal, and let $0 \ne h \in H$. Let $\mathbb{C}[z,\bar z]$ denote the complex $*$-algebra of polynomials in the coordinate function $z:\mathbb{C}\to\mathbb{C}$ and its conjugate $\bar z:\mathbb{C}\to\mathbb{C}$. Define
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\begin{align*}
H_h := \overline{\{p(T,T^*)h : p \in \mathbb{C}[z,\bar z]\}} \subset H.
\end{align*}
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Then $H_h$ is a closed reducing subspace for $T$, and if
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\begin{align*}
A := T|_{H_h} \in \mathcal{L}(H_h)
\end{align*}
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denotes the restricted normal operator, there exist a compact set $K \subset \mathbb{C}$, a finite positive regular Borel measure $\mu_h$ on $K$, and a unitary map
where $1_K:K\to\mathbb{C}$ is the constant function $1$, and $M_z \in \mathcal{L}(L^2(K,\mathcal{B}(K),\mu_h))$ is the multiplication operator defined by
for every $f\in L^2(K,\mathcal{B}(K),\mu_h)$ and for $\mu_h$-almost every $\zeta \in K$. The compact set $K$ may be taken to be $\sigma(A)=\sigma(T|_{H_h})$.