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 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 $\mathrm{span}\{e_1,\dots ,e_{m(z)}\}\subseteq {\mathrm{\ell }}^2(\mathbb{N})$, 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 $\frac{2^{-j}}{1+{\mu }_j(\sigma (T))}{\mu }_j$. (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 and whose direct sum is complete because a Cauchy sequence in the sum has summable component norms, its components converge in the complete spaces $L^2(\mu {\mathrm{\mid }}_{A_r})$, and the componentwise limit has finite norm and is the norm limit (dominated convergence for the counting measure) (Riesz-Fischer completeness of $L^p$ for $1\le p\le \mathrm{\infty }$, Dominated convergence). (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 $d{\mu }_j=h_{j}d\mu$ on $F_j$ (Integrating against a Radon-Nikodym derivative recovers integration against the measure, The total variation of an absolutely continuous signed or complex measure has density the absolute value of the Radon-Nikodym derivative); 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) The data $(\mu ,h_j,F_j,m)$ depend on the chosen decomposition; the measure class of $\mu$ and the almost-everywhere class of $m$ are independent of the choice by the intertwiner lemma and the classification theorem proved immediately below on this page. 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 $E(\{z\})$.