Invariant Subspaces for Normal Operators

(Ando) Use the following result (Corollary 9.6: if $N$ is a normal operator and $\mu$ is a scalar-valued spectral measure for $N$, then $N$ is reductive if and only if $P^\infty (\mu ) = L^\infty (\mu )$) to show that every compact normal operator is reductive.

Determine all of the invariant subspaces of a compact normal operator.

(Ando) Show that every reductive compact operator is normal. (Also see Rosenthal [1968].)

Show that an operator $A$ is reductive if and only if $\operatorname {Lat} A = \operatorname {Lat} A^*$.

Let $N$ be a normal operator on a Hilbert space $\mathcal{H}$, let $\mathcal{R}$ be an invariant subspace for $N$, and let $S = N | \mathcal{R}$. Show that $S$ is normal if and only if $\mathcal{R}$ is a reducing subspace for $N$.

Let $\mu$ be a compactly supported regular Borel measure on $\mathbb {C}$ and let $P^2(\mu )$ denote the closure in $L^2(\mu )$ of the analytic polynomials; that is, $P^2(\mu )$ is the closed linear span of $\{ z^n : n \geq 0\}$. Show that $P^2(\mu )$ is invariant for $N_\mu$ and that $N_\mu | P^2(\mu )$ is normal if and only if $P^2(\mu ) = L^2(\mu )$.

With the notation of Exercise 6, show that if $\mathcal{R}$ is a reducing subspace for $N_\mu$ and $P^2(\mu ) \subseteq \mathcal{R}$, then $\mathcal{R} = L^2(\mu )$.