Spectral multiplicity function in the separable case

Assume AC. Let $T$ be a bounded normal operator on a nonzero separable complex Hilbert space $H$ (Separability: the existence of an at most countable dense subset) with spectral projection valued measure $E$ on $\sigma (T)$ and spectral measure class to be described below.

Step 1: a countable cyclic decomposition. By the multiplication-operator form of the spectral theorem applied to a dense sequence, fix once and for all a finite or countable family $(x_j{)}_{j\in J}$ of nonzero vectors with $H={⨁}_{j\in J}{H}_{x_j},{\mu }_j:=E_{x_j}=⟨E(\cdot )x_j,x_j⟩,$ where $H_{x_j}$ is the cyclic subspace of $x_j$ and each ${\mu }_j$ is a nonzero finite positive regular Borel measure on $\sigma (T)$ (Multiplication operator form of the bounded normal spectral theorem, Cyclic vector and cyclic normal operator, Scalar and complex measures from a pvm). The index set $J$ is a finite or countable subset of $\mathbb{N}$, listed in increasing order, and summands with $x_j=0$ are omitted.

Step 2: a common dominating measure. Put $\mu :={\sum }_{j\in J}\frac{2^{-j}}{1+{\mu }_j(\sigma (T))}{\mu }_j\mathrm{.}$ Then $\mu$ is a finite positive regular Borel measure with ${\mu }_j\ll \mu$ for every $j$; let $h_j:=\frac{d{\mu }_j}{d\mu },F_j:=\{z\in \sigma (T):h_j(z)>0\}$ be the Radon–Nikodym derivative and its positivity set (A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density).

Step 3: the multiplicity function. Define the multiplicity function of the decomposition by $m(z):={\sum }_{j\in J}{\mathbf{1}}_{F_j}(z)\in \{0,1,2,\dots \}\cup \{\mathrm{\infty }\}(z\in \sigma (T))\mathrm{.}$ It is a Borel function, and $z\mapsto m(z)$ is the fiber dimension function of the model below. The set $S(z):=\{j\in J:z\in F_j\}$ of active coordinates has cardinality $m(z)$, and for $r\ge 1$ the $r$-th active coordinate is $k_r(z):=\min \{k\in J:\mathrm{\#}(S(z)\cap \{j\in J:j\le k\})=r\},z\in A_r:=\{z:m(z)\ge r\},$ a Borel function on the Borel set $A_r$; the sets $A_1\supseteq A_2\supseteq \cdots$ are decreasing and ${\bigcup }_r{A}_r$ is $\mu$-conull.

Step 4: the standard measurable-field model. The standard model of the decomposition is the space $L^2(\sigma (T),\mu ,m):=\{f=(f_r{)}_{r\ge 1}:f_r\text{ Borel},f_r=0\mu \text{-a.e. on }\{m<r\},{\sum }_{r\ge 1}\int \mathrm{\mid }{f}_r{\mathrm{\mid }}^{2}d\mu <\mathrm{\infty }\}$ of measurable fields with componentwise operations and inner product $⟨f,g⟩={\sum }_r\int f_r\overline{g_r}d\mu$ (A measurable function between measurable spaces, The space $L^p(\mu )$ as the quotient by null functions), where fields are identified when they agree $\mu$-almost everywhere in every component. Its fiber at $z$ is $\stackrel{}{\mathrm{span}}\{e_r:1\le r\le m(z)\}\subseteq {\mathrm{\ell }}^2(\mathbb{N})$, so the fiber dimension is exactly $m(z)$, and the model is the direct sum ${⨁}_{r\ge 1}{L}^2(\mu {\mathrm{\mid }}_{A_r})$ of the ordinary $L^2$-spaces of the restrictions of $\mu$ to the decreasing sets $A_r$ (Hilbert space).

Claim of the definition (identification with the operator model). The rank enumeration of active coordinates together with the Radon–Nikodym derivatives identifies the standard model unitarily with the orthogonal sum of the cyclic $L^2$-summands: the map $W:L^2(\sigma (T),\mu ,m)⟶{⨁}_{j\in J}{L}^2(\sigma (T),{\mu }_j),(Wf{)}_j(z):=\frac{f_{r(z,j)}(z)}{\sqrt{h_j(z)}}\text{ for }z\in F_j,(Wf{)}_j:=0\text{ off }{F}_j,$ where $r(z,j)$ is the rank of $j$ in $S(z)$, is a well-defined unitary operator intertwining the multiplications by every bounded Borel function and, in particular, intertwining $M_z$ with $M_z$.

Well-definedness. (1) $\mu$ is finite and nonzero and ${\mu }_j\ll \mu$ for each $j$, since the $j$-th summand of the sum dominates $\frac{2^{-j}}{1+{\mu }_j(\sigma (T))}{\mu }_j$. It is regular: for any Borel $A$ and $\epsilon >0$, choose a finite initial segment of $J$ whose weighted measure tail has total mass below $\epsilon \mathrm{/}2$. For each measure in that segment, choose a compact subset of $A$ approximating $A$ from inside, or an open superset approximating it from outside, with weighted error summing to less than $\epsilon \mathrm{/}2$. The finite union of the compact subsets lies in $A$ and the finite intersection of the open supersets contains $A$; each approximates $\mu (A)$ within $\epsilon$ because the discarded tail has total mass below $\epsilon \mathrm{/}2$. (2) $h_j$ exists, is nonnegative $\mu$-almost everywhere and is unique up to $\mu$-null sets (Radon–Nikodym); $F_j=\{h_j>0\}$ is Borel and ${\mu }_j(\sigma (T)\setminus F_j)={\int }_{\{h_j=0\}}{h}_{j}d\mu =0$. (3) ${\bigcup }_j{F}_j$ is $\mu$-conull: since $h_j=0$ off $F_j$ one has ${\mu }_j(\sigma (T)\setminus {\bigcup }_j{F}_j)={\int }_{\sigma (T)\setminus {\bigcup }_j{F}_j}{h}_{j}d\mu =0$ for every $j$, so the dominating sum vanishes there, and therefore $m\ge 1$ $\mu$-almost everywhere. (4) $m$ is Borel as a countable sum of indicators, and $k_r$ is Borel on $A_r$ because $\{k_r=k\}=F_k\cap \{{\sum }_{\mathrm{\ell }<k,\mathrm{\ell }\in J}{\mathbf{1}}_{F_{\mathrm{\ell }}}=r-1\}$ is a Borel set. (5) $L^2(\sigma (T),\mu ,m)$ is a Hilbert space: it is the orthogonal direct sum ${⨁}_r{L}^2(\mu {\mathrm{\mid }}_{A_r})$, whose components are complete by Riesz– Fischer. If $(f^{(n)})$ is Cauchy in the direct sum, each component converges in $L^2(\mu {\mathrm{\mid }}_{A_r})$ to some $f_r$. The finite partial sums of $\mathrm{\parallel }{f}_r{\mathrm{\parallel }}_2^2$ are bounded by the squared norms of the Cauchy sequence, so $(f_r)$ has finite direct-sum norm. Given $\epsilon >0$, choose $n_0$ such that $\mathrm{\parallel }{f}^{(n)}-f^{(q)}\mathrm{\parallel }\le \epsilon$ for $n,q\ge n_0$; for fixed $n\ge n_0$, pass $q\to \mathrm{\infty }$ in each finite coordinate sum and then take the supremum over those sums to get $\mathrm{\parallel }{f}^{(n)}-f\mathrm{\parallel }\le \epsilon$. Thus the direct sum is complete (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$). (6) $W$ is well defined: $(Wf{)}_j$ is Borel because on each Borel set $\{r(z,j)=r\}$ it equals $f_r\mathrm{/}\sqrt{h_j}$ with $h_j>0$ there, it vanishes off $F_j$, and it is ${\mu }_j$-square-integrable because ${\sum }_{j\in J}\int \mathrm{\mid }(Wf{)}_j{\mathrm{\mid }}^{2}d{\mu }_j={\sum }_{j\in J}{\int }_{F_j}\mathrm{\mid }{f}_{r(z,j)}{\mathrm{\mid }}^{2}d\mu ={\sum }_{r\ge 1}{\int }_{A_r}\mathrm{\mid }{f}_r{\mathrm{\mid }}^{2}d\mu =\mathrm{\parallel }f{\mathrm{\parallel }}^2<+\mathrm{\infty },$ using the positive-measure identity ${\mu }_j=h_{j}d\mu$ and the nonnegative change-of-density integral (Integrating against a density agrees with integrating the product); the same display shows $W$ is isometric, and $W$ is onto with inverse $(g_j{)}_j\mapsto f$, $f_r(z):=g_{k_r(z)}(z)\sqrt{h_{k_r(z)}(z)}$ on $A_r$ and $f_r:=0$ otherwise, which is Borel by the same rank-measurability and is square-integrable by the same computation; multiplicativity is componentwise and gives $W{M}_{\phi }=M_{\phi }W$ for every bounded Borel $\phi$, hence $W{M}_z=M_{z}W$. (7) At this stage the data $(\mu ,h_j,F_j,m)$ come from the chosen decomposition; independence of the measure class and almost-everywhere multiplicity is established later, after the intertwiner lemma. For this decomposition the multiplicity function is the fiber dimension $z\mapsto \dim H_z$ of the model, equal to $m(z)$; at a non-atomic point of $\sigma (T)$ with respect to $E$ it is not the dimension of the eigenspace $\ker (T-zI)$, which is the fiber of the atoms carried by $E(\{z\})$.