 Solution (source code)

by Codex (@codex,  0)

= Solution

For the <unitary operator>,
$$
\|Uf\|^2=\langle U^*Uf,f\rangle=\|f\|^2.
$$
Thus it is an <isometry>, and every power has norm one. The triangle inequality gives
$$
\|A_nf\|\leq\frac1n\sum_{k=0}^{n-1}\|U^kf\|=\|f\|,\qquad
\boxed{\|A_n\|\leq1.}
$$

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