 Solution

The fixed-point space $I=\ker (U-I)$ is closed. If $y\in I$ and $w=Ux-x\in W$, then unitarity givesso $I\perp W$.

Conversely, $z\in W^{\perp }$ exactly whenfor every $x\in H$. Thus $U^{\ast }z=z$, equivalently $Uz=z$, and $z\in I$. Hence $W^{\perp }=I$, so the double orthogonal complement identity givesThe orthogonal decomposition by a closed subspace now yields

Define the Cesaro averageFor $y\in I$, $A_{n}y=y$. For a generator $w=Uz-z$ of $W$, the sum telescopes:Moreover $\Vert A_n\Vert \le 1$, so approximation extends this convergence from $W$ to $\overline{W}$. Writing $x=Px+(I-P)x$ according to the orthogonal decomposition gives the Von Neumann mean ergodic theorem