Von Neumann mean ergodic theorem

For a unitary operator $U$ on a Hilbert space, the Cesaro averagesconverge strongly to the orthogonal projection onto the fixed-point subspace $\ker (U-I)$.

For a unitary operator $U$, the orthogonal complement of $\mathrm{Ran}(I-U)$ is $\ker (I-U^{\ast })=\ker (I-U)$. On the fixed space, Cesaro averages act as the identity. On $(I-U)g$ they telescope to $(g-U^{N}g)/N$. Their uniform norm bound extends convergence to the closure of the range, proving the Von Neumann mean ergodic theorem.