# Existence of an ergodic-looking limit in a Hilbert space

This is part of a problem from Reed & Simon's Functional Analysis -- I'll write the problem first.

Let $V$ be a linear transform on the Hilbert space $H$, such that its powers are uniformly bounded, i.e. $\|V^n\|<C$. Show that the operator $$U_N f= \frac{1}{N} \sum_{n=0}^{N-1} V^n f$$ converges (in norm) to the projection onto $\ker(I-V)$ as $N\to\infty$.

Let $U := \lim_{N\to\infty}U_N$; supposing $U$ exists it's a straightforward matter to verify the projection properties. So, we just need existence -- it's quick to show that $f\in\ker(I-V)\iff Uf=f=Vf$ and $f\in\mathrm{rng}(I-V)\implies Uf=0$, which may be extended to the closure of $\mathrm{rng}(I-V)$. This yields the existence of $U$ on a sizable subspace of $H$ -- however I am completely flummoxed when it comes to extending the existence to the rest of $H$. As the projection need not be orthogonal, no combination of adjoint kernels and ranges, nor arcane linear algebra techniques I can think of seem to tackle the problem after hours of scratch work; I just can't pin down the right set of properties to wrap everything up nicely. Could anyone offer some insight here? Thanks very much!

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A generalization of this question: math.stackexchange.com/q/152369 –  user48693 Nov 8 '12 at 18:44