Banach 29

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


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.


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