Star-Cyclic Normal Operators

If $\mu$ is a compactly supported measure on $\mathbb {C}$ and $f \in L^2(\mu )$, $f$ is a star-cyclic vector for $N_\mu$ if and only if $\mu (\{ x : f(x) = 0\} ) = 0$.

Prove the following proposition (Proposition 3.2).

If $\mu_1$ and $\mu_2$ are compactly supported measures on $\mathbb {C}$, show that the following statements are equivalent.

1. $\mu_1$ and $\mu_2$ are boundedly mutually absolutely continuous.

2. There is an isomorphism $V: L^2(\mu_1) \to L^2(\mu_2)$ such that $VN_{\mu_1}V^{-1} = N_{\mu_2}$ and $VL^\infty (\mu_1) = L^\infty (\mu_2)$.

3. There is a bounded bijection $R: L^2(\mu_1) \to L^2(\mu_2)$ such that $Rp(z, \bar{z}) = p(z, \bar{z})$ for every polynomial in $z$ and $\bar{z}$.

Show that if $N$ is a star-cyclic normal operator and $\lambda \in \sigma_p(N)$, then $\dim \ker (N - \lambda ) = 1$.

If $N$ is diagonalizable and star-cyclic and if $\sigma_p(N) = \{ \lambda_1, \lambda_2, \dots \}$, show that $N$ is unitarily equivalent to $N_\mu$, where $\mu = \sum_{n=1}^\infty 2^{-n} \delta_{\lambda_n}$ (see Exercise 2.11).

Let $N$ be a diagonalizable normal operator. Show that $N \cong M$ if and only if $M$ is a diagonalizable normal operator, $\sigma_p(N) = \sigma_p(M)$, and $\dim \ker (N - \lambda ) = \dim \ker (M - \lambda )$ for all $\lambda$. (Compare this with Theorem II.8.3.)

Let $U$ be the bilateral shift on $l^2(\mathbb {Z})$. If $e_0$ is the vector in $l^2(\mathbb {Z})$ that has $1$ in the zeroth place and zeros elsewhere, then $e_0$ is a star-cyclic vector for $U$. If $\mu$ is the compactly supported measure on $\mathbb {C}$ and $V: l^2(\mathbb {Z}) \to L^2(\mu )$ is the isomorphism such that $Ve_0 = 1$ and $VUV^{-1} = N_\mu$, then

Suppose $N_1, \dots , N_d$ are normal operators such that $N_j N_k^* = N_k^* N_j$ for $1 \leq j, k \leq d$ and suppose there is a vector $e_0$ in $\mathcal{H}$ such that $\mathcal{H}$ is the only subspace of $\mathcal{H}$ containing $e_0$ that reduces each of the operators $N_1, \dots , N_d$. Show that there is a compactly supported measure $\mu$ on $\mathbb {C}^d$ and an isomorphism $V: \mathcal{H} \to L^2(\mu )$ such that $VN_k V^{-1} f = z_k f$ for $f$ in $L^2(\mu )$ and $1 \leq k \leq d$ ($z_k =$ the $k$th coordinate function) (see Exercise 2.17).