Is $\|A^n\|=\|A\|^n$ true for any normal operator in a Hilbert space?

Question

Does anyone have a reference for the fact that if A is a normal operator on a Hilbert space the $\|A^n\|=\|A\|^n$? I have managed to prove this fact using a hint on another question, but I need an actual bibliographical reference, please help.

Answer 1

I believe this is even true for general C$^\ast$-algebras. If $a$ is normal in some unital C$^\ast$-algebra $A$, then $\Vert a \Vert = r(a)$ with $r(a)$being the spectral radius of $a$. Now, $r(a^n)=r(a)^n$ ensures that $$\Vert a^n \Vert = r(a^n) = r(a)^n = \Vert a \Vert^n$$

As such, I think authors tend to avoid deriving this identity, perhaps using it as an exercise in spectral theory. If you want a reference for the identity $r(a^n)=r(a)^n$, I would say theorem 5.4 in Kehe Zhu's book "An Introduction to Operator Algebras" does the job.