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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.

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    $\begingroup$ Almost any book on Functional Analysis will give the detail that you want, from Walter Rudin to Peter Lax. $\endgroup$ – DisintegratingByParts Mar 13 '17 at 20:32
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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 identify, 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.

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