Polynomial evaluated on a Normal Bounded Linear Operator

Question

Let X be a complex Hilbert Space and A be a Normal Bounded Linear Operator.

Show that the radius of the spectrum of A is equal to the norm of A.

Deduce that if P is a polynomial, then the norm of P(A) is equal to the supremum, over complex numbers z in the spectrum of A, of |P(z)|.

I can do the first, but not the second part of this question !! Any help would be much appreciated.

Answer 1

If $P$ is a polynomial, then $P(A)$ is normal, assuming $A$ is normal. Therefore, $\|P(A)\| = r_{\sigma}(P(A))$ is the spectral radius of $P(A)$. But the spectrum of $P(A)$ is $P(\sigma(A))$ by the spectral mapping theorem. Therefore, $$ \|P(A)\|=\sup_{\lambda\in\sigma(A)}|P(\lambda)|. $$