Orthogonal decomposition for unitary ergodic averages ($\mathcal{H}=\ker (I-U)\oplus \overline{\mathrm{Ran}(I-U)}$)

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.