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

The important fact about this factor is the following

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<f<1} 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

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About Joel Moreira

PhD Student at OSU in Mathematics. I'm portuguese.
This entry was posted in Ergodic Theory and tagged , , , . Bookmark the permalink.

3 Responses to Koopman-von Neumann Decomposition

  1. Pingback: Proof of Roth’s Theorem using Ergodic Theory | YAMB

  2. Pingback: Weak Mixing | YAMB

  3. Pingback: Szemerédi’s Theorem Part II – Overview of the proof | I Can't Believe It's Not Random!

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