Orthogonal decomposition for unitary ergodic averages
= Orthogonal decomposition for unitary ergodic averages
{title2=$\mathcal H=\ker(I-U)\oplus\overline{\operatorname{Ran}(I-U)}$}
For a <unitary operator> $U$, the orthogonal complement of $\operatorname{Ran}(I-U)$ is $\ker(I-U^*)=\ker(I-U)$. On the fixed space, <Cesàro averages> act as the identity. On $(I-U)g$ they telescope to $(g-U^Ng)/N$. Their uniform norm bound extends convergence to the closure of the range, proving the <Von Neumann mean ergodic theorem>.