Multiplicity Theory for Normal Operators: A Complete Set of Unitary Invariants

Let $A$ and $B$ be operators on $\mathcal{H}$ and $\mathcal{K}$, respectively. Let $\mathcal{H}_0$ and $\mathcal{K}_0$ be reducing subspaces for $A$ and $B$ and suppose that $A \cong B | \mathcal{K}_0$ and $B \cong A | \mathcal{H}_0$. Show that $A \cong B$.

Let $\mu_1, \mu_2, \dots$ be compactly supported measures on $\mathbb {C}$ such that $\mu_{n+1} \ll \mu_n$ for all $n$. Show that if $M$ is any normal operator whose spectral measure is absolutely continuous with respect to each $\mu_n$, then

If $\mu$ is Lebesgue measure on $[0, 1]$, show that $N_\mu \cong N_\mu^p$ for $0 < p < \infty$.

Let $\mu$ be Lebesgue measure on $[0, 1]$ and characterize the functions $\phi$ in $L^\infty (\mu )$ such that $N_\mu \cong \phi (N_\mu )$.

Let $\mu$ be area measure on $\mathbb {D}$ and show that $N_\mu$ and $N_\mu^2$ are not unitarily equivalent.

Let $\mu$ be Lebesgue measure on $[0, 1]$ and let $\nu$ be Lebesgue measure on $[-1, 1]$. Show that $N_\nu^2 \cong N_\mu \oplus N_\mu$. How about $N_\nu^3$?

Let $\mu$ be Lebesgue measure on $\mathbb {R}$ and $N =$ multiplication by $\sin x$ on $L^2(\mu )$. Find the decompositions of $N$ obtained in Theorem 10.1 and Theorem 10.16, stated below.

Theorem 10.1: (a) If $N$ is a normal operator, then there is a sequence (possibly finite) of measures $\{ \mu_n\}$ on $\mathbb {C}$ such that $\mu_{n+1} \ll \mu_n$ for all $n$ and $N \cong N_{\mu_1} \oplus N_{\mu_2} \oplus \cdots$. (b) If $N$ and $\{ \mu_n\}$ are as in (a) and $M \cong N_{\nu_1} \oplus N_{\nu_2} \oplus \cdots$, where $\nu_{n+1} \ll \nu_n$ for all $n$, then $N \cong M$ if and only if $[\mu_n] = [\nu_n]$ for all $n$.

Theorem 10.16: If $N$ is a normal operator, then there are mutually singular measures $\mu_\infty , \mu_1, \mu_2, \dots$ (some of which may be zero) such that

If $M$ is another normal operator with corresponding measures $\nu_\infty , \nu_1, \nu_2, \dots$, then $N \cong M$ if and only if $[\mu_n] = [\nu_n]$ for $1 \leq n \leq \infty$.

If $\mu$ is Lebesgue measure on $\mathbb {R}$ and $N =$ multiplication by $e^{ix}$ on $L^2(\mu )$, show that $N \cong N_m^{(\infty )}$, where $m =$ arc length measure on $\partial \mathbb {D}$.

Define $U: L^2(\mathbb {R}) \to L^2(\mathbb {R})$ by $(Uf)(t) = f(t - 1)$. Show that $U$ is unitary and find its scalar-valued spectral measure and multiplicity function.

Represent $N$ as in the following theorem (Theorem 10.1(a): if $N$ is a normal operator, then there is a sequence (possibly finite) of measures $\{ \mu_n\}$ on $\mathbb {C}$ such that $\mu_{n+1} \ll \mu_n$ for all $n$ and $N \cong N_{\mu_1} \oplus N_{\mu_2} \oplus \cdots$) and find the corresponding representation for $N \oplus N = N^{(2)}$; for $N^{(3)}$; for $N^{(\infty )}$. (Are you surprised by the result for $N^{(\infty )}$?)

Prove the results and solve the exercises from II.8.

Let $N$ be a normal operator and show that $N \cong N^{(2)}$ if and only if there is a $*$-cyclic normal operator $M$ such that $N \cong M^{(\infty )}$. What does this say about the multiplicity function for $N$?

Let $(X, \Omega , \mu )$ be a measure space such that $L^2(\mu )$ is separable, let $\phi \in L^\infty (\mu )$, and let $N = M_\phi$ on $L^2(\mu )$. Find the decompositions of $N$ obtained in Theorem 10.1 and Theorem 10.16, stated below.

Theorem 10.1: (a) If $N$ is a normal operator, then there is a sequence (possibly finite) of measures $\{ \mu_n\}$ on $\mathbb {C}$ such that $\mu_{n+1} \ll \mu_n$ for all $n$ and $N \cong N_{\mu_1} \oplus N_{\mu_2} \oplus \cdots$. (b) If $N$ and $\{ \mu_n\}$ are as in (a) and $M \cong N_{\nu_1} \oplus N_{\nu_2} \oplus \cdots$, where $\nu_{n+1} \ll \nu_n$ for all $n$, then $N \cong M$ if and only if $[\mu_n] = [\nu_n]$ for all $n$.

Theorem 10.16: If $N$ is a normal operator, then there are mutually singular measures $\mu_\infty , \mu_1, \mu_2, \dots$ (some of which may be zero) such that

If $M$ is another normal operator with corresponding measures $\nu_\infty , \nu_1, \nu_2, \dots$, then $N \cong M$ if and only if $[\mu_n] = [\nu_n]$ for $1 \leq n \leq \infty$.

Let $\mu$ be a compactly supported measure on $\mathbb {C}$, $\phi$ a bounded Borel function on $\mathbb {C}$, and suppose $\{ \Delta_n\}$ are pairwise disjoint Borel sets such that $\phi$ is one-to-one on each $\Delta_n$ and $\mu (\mathbb {C} \setminus \bigcup_{n=1}^\infty \Delta_n) = 0$. Let $\phi_n = \phi \chi_{\Delta_n}$ and $\mu_n = \mu \circ \phi_n^{-1}$ for $n \geq 1$. Prove that $M_\phi$ on $L^2(\mu )$ is unitarily equivalent to $\bigoplus_{n=1}^\infty N_{\mu_n}$.