Riesz-Markov to define the Functional Caluculus, and the measure of the spectrum.

Question

Let $H$ be a Hilbert space. Let $A$ be self-adjoint, and $f(A)$ denotes the continuous functional calculus.

Let $\psi \in H$. Is it true that for $T_\psi: C(\sigma(A)) \to \mathbb{C}$ given by $T_\psi(f) = (\psi, f(A)\psi)$ we have that $||T_\psi|| = ||\psi||^2$?

One inequality is obvious, $||T_\psi|| \geq ||\psi||$ using $f = 1$.

Help with the other, if true?

Answer 1

Actually :):

By C.S and the continuous functional calculus:

$sup_{||f||_\infty=1}(\psi, f(A)\psi) \leq sup_{||f||_\infty=1}||\psi||$ $||f(A)\psi|| \leq sup_{||f||_\infty=1}||\psi||^2||f|| = ||\psi||^2$