The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$, where ${\cal H}$ is a Hilbert space, the following holds: $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n v = \Pi_U v, $$ where $\Pi_U$ is the orthogonal projection onto the invariant subspace of $U$.

Following answers by Coudy and Asaf, and comment by Terry Tao:

Suppose one wishes to describe the "typical" behavior of the convergence rate, say, how close does the sum vector $\frac{1}{N}\sum_{n=1}^N U^n v$ get to the invariant subspace of $U$ in terms of $l_2$-norm.

In general the convergence rate depends heavily on the unitary $U$, and in some cases it is not even computable (see Tao's comment on metastability). Hence there is no "typical" behavior to speak of.

Would it be then reasonable to conjecture a "density"-type argument as follows: For every $U$, there exists a "nearby" $U'$, $\|U - U'\| = o(1)$, such that $U'$ convergence rate is at least, say inverse-polynomial in $dim({\cal H})$ ?

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    $\begingroup$ If one wants quantitative forms of the mean ergodic theorem for arbitrary shift operators $U$, then the correct question to ask is not to bound the convergence rate, but rather to bound the metastability of the averages. See e.g. andrew.cmu.edu/user/avigad/Talks/dc.pdf . The prototypical result in which metastability is quantitative but convergence rate is not is the basic real analysis fact that all bounded monotone sequences converge: see terrytao.wordpress.com/2007/05/23/… . $\endgroup$ – Terry Tao May 15 '15 at 15:33

It depends on $U$ of course. If $U$ is the identity, the convergence is pretty fast. It also depends on $f$. If $f$ is a coboundary ($f=g-Ug$), then the convergence is of the order $1/n$.

When $U$ comes from a dynamical system, there are a number of results showing that there is usually no rate of convergence in the ergodic theorem. These theorems are stated as counterexamples to the a.e. convergence. Let me just mention one result from the book of Petersen, "Ergodic Theory" (3.2 B).

Theorem (Kakutani-Petersen) Let $T$ an ergodic measure transformation of a non-atomic probability space and $(b_i)$ a positive sequence with divergent sum. Then there is a bounded function $f$ with zero integral such that $$ sup_N \left|\sum_{n=1}^N b_n {1\over n} S_nf(x) \right| = \infty \quad for\ a.e. \ x. $$ denoting as usual $Uf = f\circ T$ and $S_Nf = {1\over N} \sum_{k=1}^NU^kf$.

This implies that $\sum b_n \| {1\over n} S_nf\|_2$ is divergent and shows that for any positive sequence $(a_n)$ going to zero, there can't be a bound $\| {1\over n} S_nf\|_2 \leq a_n$ (choose $b_n$ such that $\sum b_n = \infty$ and $\sum b_n a_n < \infty$).

In contrast, assume that the sequence $(f\circ T^n)$ is independent, then there is a convergence rate given by the iterated law of the logarithm or the Kolmogorov three-series theorem.


In general, it varies. There are cases where the convergence is quite fast (for example in the case where the system is mixing, and say in the presence of spectral gap, think of Bernoulli system or say geodesic flow on $PSL_{2}(\mathbb{R})/\Gamma$).

On the other hand, the convergence can be rather slow in Kronecker systems (hence also in the Kronecker factor of your system, which will appear in the Hilbert space $L^{2}(\mu)$ unless your system is weak-mixing). The reason is simple, you need to estimate the exponential sum $\frac{1}{N}\sum_{n=0}^{N-1}e_{\alpha}(n)$ where $e_{\alpha}(x)=exp(2\pi i \alpha x)$ and $\alpha$ is the corresponding eigenvalue. In the case that $\alpha$ is Liouvillian the decay rate of this sum can be rather bad. Somehow the most precise results appear in the famous Green-Tao paper about nilflows - http://arxiv.org/abs/0709.3562 .

ADDED IN EDIT - just a few remarks over the first part of my answer. Even in the mixing case (think about Bernoulli system, or horocycle flow), is that you have some sort of shift operator (this is not entirely correct in the horocyclic case, I'll correct that later), and then the mixing statement (which implies the mean ergodic theorem) is basically equivalent to the fact that $\lim_{t\to\infty} \int_{X}f_{1}(T_{t}x)f_{2}(x)d\mu(x) = \int f_{1} \int f_{2}$, where $T_{t}$ is some shifting operator and $f_{i}\in L^{2}(\mu)$ (you should think about the Bernoulli case here). In general, unless you limit your functions in consideration to a nice set of functions (say certain decay or smoothness or limit their support), you won't be able to gain anything substantial (think about the Bernoulli case, with two long cylindrical functions, where you will need to shift for quite a lot of time to make them independent, and you may take the cylinder to be as long as you may like). Therefore many of those mixing results would be true only for certain class of functions (Sobolov is quite a common choice, Lipschitz usually suffices). Regarding the horocyclic flow - by taking the "non compact line model of a principal series representation of $SL_{2}(\mathbb{R})$", whatever that means, you can find a nice "portion" (actually, a prototypical direct summand) in $L^{2}_{0}(G/\Gamma)$ which will be isometric (and actually, infinitesimally equivalent to) $L^{2}(\mathbb{R})$ and the action of $U_{t}$ is equivalent to shifting $U_{t}.f(x)=f(x-t)$. This shows that even in the nicest mixing system one might possibly imagine, one can get vectors for which the correlation decay would be as slow as you want (and as the system is mixing, the Kronecker factor is trivial, there are no non-constant eigenfunction to play with, in contrast to the situation I described above).


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