Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 1 / 5 / d / Solution

Put $F=\ker (I-U)$. For a unitary operator, $\ker (I-U^{\ast })=F$, soThis follows from the general identity $\mathrm{Ran}(A{)}^{\perp }=\ker A^{\ast }$. The averaging operators have $\Vert T_n\Vert \le 1$ and equal the identity on $F$. On a vector $(I-U)\eta$, telescoping givesThe uniform norm bound extends this convergence to the closure of the range. Thus $T_n$ tends to identity on $F$ and to zero on its orthogonal complement. Consequentlywhich is exactly convergence in the strong operator topology. This proves the Von Neumann mean ergodic theorem in the requested unitary case.