It's in Nagy's book "Harmonic Analysis of Operators on Hilbert space" on page 31, the proof of the von Neumann inequality for polynomials.
What is the "spectral representation of unitary operators"? And why does it imply that $\| p (U) \|$ is equal to maximum of $| p(\lambda) |$ on the spectrum of $U$ (where $U$ is a unitary operator, and $p$ a polynomial of degree $n$)?
I tried to find more about this, but couldn't.