# Banach 29

This lecture (banach29 ;don’t mind the title) was given on 11/2/09

### Excerpts

$L^\infty$ functional calculus.

$T \in \mathscr{L}(H)$ normal, for $x \in H$, $M_x$ is reducing subpsace of $T$ gen. by $x$.  Know $T|M_x$ is unit. equiv to $T_{\mu_x}$, $\mu_x$ a Borel measure.  For $\phi \colon \mathbb{C} \to \mathbb{C}$ a bounded Borel measurable function can make sense of $\phi(T)$ via

$\phi(T) = \sum \oplus \phi(T|_{M_x})$

its well defined…

Review Rmk and

Lemma III.3.1 $\phi$ as above, $x \in H - 0$ then $\langle \phi(T)x,x\rangle = \int \phi d\mu_x$.

Lemma III.3.2 If $x \in H - 0$ and if $E \subset \mathbb{C}$ is a Borel set then $\mu_x(E) = 0$ iff $\chi_E(T)x = 0$.

$\mu << \nu$ if $\nu(E) = 0$ implies $\mu(E) = 0$.

Prop. III.3.3 $\exists x' \in H$ s.t. $\mu_x << \mu_{x'}$ for all $x \in H$

Such a measure $\mu_{x'}$ is called a scalar spectral measure for $T$.

and there’s another contstruction $S \mapsto E(S)$ that I should understand better.