Continuous functional calculus

Question

Let $\mathscr H$ be a Hilbert space, and $\mathscr B(\mathscr H)$ is a $C^*$-algebra, $T\in \mathscr B(\mathscr H)$ is a normal operator. Let $C^*(T)$ denote the $C^*$- subalgebra generated by $T$ and $I$. For any function $f\in C_\infty(\sigma(T))$ we can define functional calculus $f(T)$ to be the inverse image of $f$ under the Gelfand tranform. Let $\xi$ be any element in $\mathscr H$. My question is how to calculate $f(T)\xi$ ?

Answer 1

It is clear when $f$ is a polinomial function, indeed $f(T)\xi = \sum_1^N \alpha_iT^i\xi$. When $f$ is a general $C_{\infty}$ function you can try to say something using Weierstrass approximation theorem.

I know it's not very much, but I wish it to be of some help.