# Banach 25

This lecture ( Banach25) was given on 10/23/09.

### Excerpts

Definitions

1. $\mu$ a positive, finite, compactly supported Borel measure on $\mathbb{C}$, then define $T_\mu \in \mathscr{L}(L^2(\mu))$ via $(T_\mu f)(z) = zf(z)$.  $T_\mu$ is normal and called a canonical multiplication operator.
2. $T \in \mathscr{L}(H)$ is cyclic if $\exists x$ such that images of $x$ under polynomials in $T,T^*$ are dense in $H$, in this case $x$ is a cyclic vector for $T$ (e.g. $T = T_\mu$ and $x = 1$)
3. $T_i \in \mathscr{L}(H_i)$ are unitarily equivalent, if $\exists U \in \mathscr{L}(H_2,H_1)$ that is unitary such that $T_2 = U^{-1}TU$.

Results

1. (Spectral Thm for compact normal operators) $T \in \mathscr{L}(H)$ compt. normal.  Let $\lambda_1, \lambda_2,...$ be its nonzero eigenvalues and $M_1, M_2, ...$ corresponding eigenspaces; $M_0 = \ker T$.  Then $H = \sum_{n\ge 0} \oplus M_n$ (internal direct sum).
2. Thm III.3.1 Every cyclic normal operator is unitarily equivalent to a canonical multiplication operator.