## Koopman-von Neumann Decomposition

In my previous post I presented an ergodic theoretical proof of Roth’s Theorem, assuming the Koopman-von Neumann Decomposition (and some other minor facts). In this post I present a proof of this Decomposition and moreover prove that the compact vectors form a factor.

— 1. Introduction —

Recall that a measure preserving system is a quadruple ${(X,{\cal B},\mu,T)}$ where ${(X,{\cal B})}$ is a Borel space, ${\mu}$ is a probability measure and ${T:X\rightarrow X}$ is an invertible measure preserving map. We denote by ${H}$ the Hilbert space of all ${L^2}$ functions on ${(X,\mu)}$ and define the unitary operator ${U:H\rightarrow H}$ by ${(Uf)(x):=f(Tx)}$. In this post we are interested in writing ${H}$ as the direct sum of two orthogonal subspaces, one contains the compact vectors and the other contains the weak-mixing vectors. We defined these objects now:

Definition 1 Let ${f\in H}$. We say that ${f}$ is a compact or almost periodic function if the orbit closure ${\overline{\{U^nf|n\in{\mathbb Z}\}}\subset H}$ is compact as a subset of ${H}$ with the strong topology. The set of all compact functions is denoted by ${H_c}$.

Definition 2 Let ${g\in H}$. We say that ${g}$ is a weak-mixing function if

$\displaystyle \lim_{n\rightarrow\infty}\frac1n\sum_{k=1}^n|\langle U^kg,g\rangle|=0$

The set of all weak-mixing functions is denoted by ${H_{wm}}$.

We can now state the Koopman-von Neumann Decomposition:

Theorem 3 (Koopman-von Neumann) The set ${H_c}$ is a closed subspace of ${H}$ and the it’s orthogonal complement is ${H_{wm}}$.

— 2. Proof of the Koopman-von Neumann Decomposition —

We separate the proof into some lemmas:

Lemma 4 Let ${\phi:H^2\rightarrow H}$ be a function that commutes with ${U}$ (so that ${\phi(Uf,Ug)=U\phi(f,g)}$) and is uniformly continuous.

Then that for ${f,g\in H_c}$ also ${\phi(f,g)\in H_c}$.

We remark that, for instance, the function ${\phi(f,g)=f+g}$ satisfies the hypothesis.

Proof:

Let ${f,g\in H_c}$ and fix ${\epsilon>0}$. Let ${\delta>0}$ be such that if ${f',f'',g',g''\in H}$ are such that ${\|f'-f''\|<\delta}$ and ${\|g'-g''\|<\delta}$ then ${\|\phi(f',g')-\phi(f'',g'')\|<\epsilon}$.

Also let ${\{B_i\}}$ and ${\{C_i\}}$ be finite covers of the orbit closure of ${f}$ and ${g}$ (respectively) by balls with diameter less than ${\delta}$. Then for any ${n\in{\mathbb Z}}$ we have ${U^nf\in B_i}$ and ${U^ng\in C_j}$ for some ${i,j}$ depending on ${n}$. Therefore ${U^n\phi(f,g)=\phi(U^nf,U^ng)\in\phi(B_i,C_j)}$ and by construction ${\phi(B_i,C_j)}$ has diameter at most ${\epsilon}$. This implies that the orbit of ${\phi(f,g)}$ is contained in the union of finitely many sets with diameter ${\epsilon}$ (namely ${\{\phi(B_i,C_j)\}_{i,j}}$) and therefore is totaly bounded. Hence ${\phi(f,g)}$ is a compact function. $\Box$

Now I will state a fact about weak-mixing functions. In the proof I will need the van der Corput Trick that I proved in my previous post.

Proposition 5 Let ${f\in H_{wm}}$ and ${g\in H}$. Then

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N|\langle U^nf,g\rangle|=0$

Proof:

Let ${u_n=\langle U^nf,g\rangle U^nf}$. We have

$\displaystyle \begin{array}{rcl} &&\displaystyle\lim_{H\rightarrow\infty}\frac1H\sum_{h=1}^H\lim_{N\rightarrow\infty}\frac1N\sum_{n=0}^{N-1} \langle u_{n+h},u_n\rangle\\&=&\displaystyle\lim_{H\rightarrow\infty}\frac1H\sum_{h=1}^H\lim_{N\rightarrow\infty}\frac1N\sum_{n=0}^{N-1} \langle U^{n+h}f,g\rangle\overline{\langle U^nf,g\rangle}\langle U^{n+h}f^,U^nf\rangle\\&\leq&\displaystyle \lim_{H\rightarrow\infty}\frac1H\sum_{h=1}^H\|f\|^2\|g\|^2|\langle U^hf,f\rangle|\\&=&0 \end{array}$

By the van der Corput Trick we conclude that

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=0}^{N-1}\langle U^nf,g\rangle U^nf=0$

and thus

$\displaystyle \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N|\langle U^nf,g\rangle|^2= \lim_{N\rightarrow\infty}\left\langle\frac1N\sum_{n=1}^N\langle U^nf,g\rangle U^nf,g\right\rangle =0$

$\Box$

We can now prove that ${H_c}$ and ${H_{wm}}$ are orthogonal sets:

Lemma 6 Let ${f\in H_{wm}}$ and ${g\in H_c}$. Then ${\langle f,g\rangle=0}$.

Proof:

Fix ${\epsilon>0}$. Let ${g_1,...,g_n}$ be such that the ball ${B(g_i,\epsilon)}$ cover the orbit of ${g}$. Thus for a fixed ${m}$ let ${i}$ be such that ${U^m g\in B(g_i,\epsilon)}$. We have

$\displaystyle |\langle f,g\rangle|=|\langle U^m f, U^mg\rangle|\leq \epsilon\|f\|+|\langle U^mf,g_i\rangle|\leq\epsilon\|f\|+\sum_{i=1}^n|\langle U^mf,g_i\rangle|$

Thus we have

$\displaystyle |\langle f,g\rangle|=\frac1M\sum_{m=1}^M|\langle U^mf,U^mg\rangle|\leq\epsilon\|f\|+ \sum_{i=1}^n\frac1M\sum_{m=1}^M|\langle U^mf,g_i\rangle|$

Since ${f\in H_{wm}}$, using lema 5 with for ${M}$ large enough we have that the second term can be made smaller than ${\epsilon\|f\|}$. Thus we have ${|\langle f,g\rangle|\leq 2\epsilon\|f\|}$ and since ${\epsilon>0}$ was arbitrary we conclude that actually ${\langle f,g\rangle=0}$. $\Box$

We now need a converse of the previous lemma:

Lemma 7 Let ${g\in H}$ be not weak mixing. Then there exist some ${f\in H_c}$ such that ${\langle f,g\rangle\neq0}$.

Proof: Consider the operator ${\phi_g:f\mapsto\langle f,g\rangle g}$. This is rank one and thus a Hilbert-Schmidt operator. In the Hilbert space ${HS}$ of all Hilbert-Schmidt operators on ${H}$ (with the Hilbert-Schmidt norm) define the unitary operator ${V:HS\rightarrow H_S}$ by ${V(\psi)=U\psi U^{-1}}$. Thus ${V(\phi_g)=\phi_{Ug}}$.

By the von Neumann’s Mean Ergodic Theorem, we have that

$\displaystyle \psi_g:=\lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N V^n\phi_g$

exists, is in ${HS}$ and is invariant under ${V}$. In other words ${\psi_g}$ commutes with ${U}$.

We will prove that ${f=\psi_gg}$ satisfies the claims. Note that ${\langle (V^n\phi_g)g,g\rangle=|\langle g,U^ng\rangle|^2}$ so

$\displaystyle \langle f,g\rangle=\left\langle\lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N (V^n\phi_g)g,g\right\rangle = \lim_{N\rightarrow\infty}\frac1N\sum_{n=1}^N|\langle g,U^ng\rangle|^2>0$

The last inequality is from the definition of weak mixing function (which ${g}$ is not) and we can pass the limit outside the inner product because convergence in the Hilbert-Schmidt norm implies convergence in the weak operator topology.

Now all that remains to prove is that ${f\in H_c}$. But since ${\psi_g}$ is an Hilbert-Schmidt operator (and hence compact) and commutes with ${U}$, the orbit of ${f=\psi_gg}$ (under ${U}$) is the image under ${\psi_g}$ of the orbit of ${g}$. But since ${U}$ is unitary, the orbit of ${g}$ is bounded and therefore the orbit of ${f}$ is pre-compact. In other words, ${f}$ is a compact vector.

$\Box$

We are now ready to prove the Koopman-von Neumann Decomposition

Proof: of Theorem 3

First note that ${H_c}$ is a ${U}$-invariant set, because ${f}$ and ${Uf}$ have the same orbit, and it’s closed under scalar multiplication, because the scalar multiple of a compact set is still compact.

We claim that ${H_c}$ is a closed set because the orbit of a limit point of functions in ${H_c}$ has a totaly bounded (and hence pre-compact) orbit. Indeed let ${f\in\overline{H_c}}$ and fix ${\epsilon>0}$. Choose ${g\in H_c}$ such that ${\|f-g\|<\epsilon/3}$. Since ${g}$ is compact there exists a finite set ${A\subset{\mathbb Z}}$ such that the balls ${B(U^ag,\epsilon/3)}$ cover the orbit of ${g}$. For each ${n\in{\mathbb Z}}$ let ${a\in A}$ be such that ${\|U^ag-U^ng\|<\epsilon/3}$. Then

$\displaystyle \|U^af-U^nf\|\leq \|U^af-U^ag\|+\|U^ag-U^ng\|+\|U^ng-U^nf\|=2\|f-g\|+\|U^ag-U^ng\|<\epsilon$

Now by Lemma 4 we have that ${H_c}$ is closed under addition and hence is a closed invariant subspace of ${H}$. By Lemma’s 6 and 7 we conclude that ${f\in H_{wm}}$ if and only if it is orthogonal to ${H_c}$, hence ${H_{wm}=H_c^\perp}$ and this concludes the proof. $\Box$

— 3. The Kronecker Factor —

Given a measure preserving system ${(X,{\cal B},\mu,T)}$ we call a factor to a sub-${\sigma}$-algebra ${{\cal C}}$ of ${{\cal B}}$ which is invariant under ${T}$, i.e., such that for any ${A\in{\cal C}}$ we have that ${T^{-1}A\in{\cal C}}$ as well.

Proposition 8 (Kronecker factor) The family ${{\cal K}}$ consisting of all sets ${A\in{\cal B}}$ such that ${1_A\in H_c}$ is a factor. We call ${{\cal K}}$ the Kronecker factor of ${(X,{\cal B},\mu,T)}$.

Proof:

Since ${H_c}$ is an invariant subspace of ${H}$ we have that ${{\cal K}}$ is an invariant family of subsets. Also since the constant function ${1}$ is in ${H_c}$ (and it is closed for sums) we get that ${{\cal K}}$ is invariant under complements.

Since the function ${\phi:H^2\rightarrow H}$ given by ${\phi(f,g)=\max(f,g)}$ is uniformly continuous, by Lemma 4 and the Monotone Convergence Theorem we get that ${{\cal K}}$ is also closed under countable unions, so it’s indeed a ${\sigma}$-algebra. $\Box$

Proposition 9 Let ${f\in H}$. Then ${f\in H_c}$ if and only if it is measurable with respect to the Kronecker factor ${{\cal K}}$.

Proof:

If ${f}$ is measurable with respect to ${{\cal K}}$ then ${f}$ can be approximated by simple functions in ${H_c}$. Since ${H_c}$ is a closed subspace also ${f\in H_c}$.

Reciprocally if ${f\in H_c}$, then for each ${a\in {\mathbb R}}$ we have, by Lemma 4, that ${\max(f,a)\in H_c}$, so also ${n[\max(f,a)-a]\in H_c}$ for each ${n\in{\mathbb N}}$ and so ${\min(1,n[\max(f,a)-a])\in H_c}$. Taking the limit when ${n\rightarrow\infty}$ and using the Dominated Convergence Theorem we conclude that the characteristic function of ${\{f>a\}}$ is in ${H_c}$ and so ${f}$ is measurable with respect to ${{\cal K}}$ as desired. $\Box$

An immediate Corollary of this is this Proposition left unproved in my last post.

As another application of Lemma 4 I now finish the proof of Roth’s Theorem (which in my last post I showed follows from the following theorem), using two key Propositions from my last post:

Theorem 10 (Roth’s Theorem, ergodic version) Let ${(X,{\cal B},\mu,T)}$ be an ergodic measure preserving system and let ${A\in{\cal B}}$ have positive measure. Then there exists ${n\in{\mathbb N}}$ such that ${\mu(A\cap T^{-n}A\cap T^{-2n}A)>0}$.

Proof: Let ${1_A}$ be the characteristic function of ${A}$ and using Theorem 3 split it as ${1_A=f+g}$ where ${f\in H_c}$ and ${g\in H_{wm}}$. Note that ${0 because otherwise the function ${\max(\min(f,1),0)}$ is also in ${H_c}$ (by Lemma 4) and would be closer to ${1_A}$ then ${f}$. Moreover, representing the constant function equal to ${1}$ (which is compact and thus orthogonal to ${g}$) simply by ${1}$ we have

$\displaystyle \int_Xfd\mu=\langle f,1\rangle=\langle 1_A,1\rangle=\int_X1_Ad\mu=\mu(A)>0$

We showed that ${f}$ is in the conditions of Lemma 13 in my previous post. Moreover, by Lemma 12 in that post, we conclude that

$\displaystyle \liminf_{N\rightarrow\infty}\frac1N\sum_{n=1}^{N-1}\mu(A\cap T^{-n}A\cap T^{-2n}A)>0$

Therefore there is at least on ${n}$ such that ${\mu(A\cap T^{-n}A\cap T^{-2n}A)>0}$. $\Box$