Posted by: yglima | June 11, 2010

ERT12: Kronecker factor – coexistence of compact and weak mixing behaviour

We have studied two behaviors that are, in the spectral sense, disjoint. It turns out that they are also complementary, meaning that every measure-preserving system can be decomposed in a component which behaves in a compact fashion and its complement in a weakly mixing way. The compact component defines a {\sigma}-algebra {\mathcal Z_1} known as the Kronecker factor, which encapsulates all the linear structure of the system. Actually, the next post will prove this component is enough to obtain the case {k=3} of Furstenberg theorem: this is known as Roth theorem, who first proved the existence of arithmetic progressions of length 3 in subsets of {\mathbb Z} with positive density.

As we saw in ERT8 and ERT10, weak mixing and compact systems are identified by verifying numerical and topological properties of the sequences {(T^nf)_{n\in\mathbb Z}}. Here, {T} stands for the {\mathbb Z}-action defined by the mps {\mathbb X=(X,\mathcal A,\mu,T)} and also for the Koopman-von Neumann operator {T:L^2(\mu)\rightarrow L^2(\mu)}. More specifically, {\mathbb X} is weak mixing iff

\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|(T^nf,f)\right|=0\,,

for any {f\in L^2(\mu)} such that {\int_Xfd\mu=0}, and compact iff \displaystyle \{T^nf;n\in\mathbb Z\} is pre-compact in L^2(\mu), for any {f\in L^2(\mu)}.

Yet considering the spectral point of view, let {\mathcal H} be a Hilbert space and {T:\mathcal H\rightarrow\mathcal H} a unitary operator. Motivated by the above situation, let us make the following

Definition 1 An element {x\in\mathcal H} is weak mixing if\displaystyle \lim_{N\rightarrow+\infty}\dfrac{1}{N}\sum_{n=1}^N\left|(T^nx,x)\right|=0

and almost periodic if the set {\{T^nx;n\in\mathbb Z\}} is pre-compact in {\mathcal H}.

Consider the sets

\displaystyle \begin{array}{rcl} \mathcal H_c&=&\{x\in\mathcal H\,;\,x\text{ is almost periodic}\}\\ \mathcal H_{wm}&=&\{x\in\mathcal H\,;\,x\text{ is weak mixing}\}. \end{array}

The main theorem of this post, due to Koopman and von Neumann, estabilishes the equality {\mathcal H=\mathcal H_c\oplus^\perp\mathcal H_{wm}}. In order to do this, we characterize {\mathcal H_c} and {\mathcal H_{wm}} in different manners.

Proposition 2 {\mathcal H_c} is equal to the closure of the union of eigenspaces:\displaystyle \mathcal H=\overline{\bigoplus_{\lambda\in\sigma(T)}\{x\in\mathcal H\,;\,Tx=\lambda x\}}\,.

Proof: The inclusion {\subset} is clear: if {Tx=\lambda x}, the closure of {\{T^nx;n\in\mathbb Z\}} is equal to {\Lambda x}, where {\Lambda} is a subgroup of {\mathbb S^1}. To the reverse inclusion, consider {x\in\mathcal H} almost periodic and let {A=\overline{\{T^nx;n\in\mathbb Z\}}}. The restriction map {T|_A:A\rightarrow A} is a compact operator. Being also self-adjoint, the spectral theorem implies the existence of an orthonormal basis of eigenvectors of {T|_A}. In particular, {x} is in the closure of the subspace generated by them, which concludes the proof. \Box

The alternative characterization of {\mathcal H_{wm}} uses the van der Corput trick (see ERT9). Let us remember it.

Theorem 3 (van der Corput trick) If {(x_n)_{n\in{\mathbb N}}} is a bounded sequence in a Hilbert space {\mathcal H} and if

\displaystyle \lim_{H\rightarrow+\infty}\limsup_{N\rightarrow+\infty}\dfrac{1}{H}\sum_{h=1}^H\dfrac{1}{N}\sum_{n=1}^N (x_{n+h},x_n)=0\,, \ \ \ \ \ (1)

then {\left\|\dfrac{1}{N}\sum_{n=1}^N x_n\right\|\rightarrow 0}.

Proposition 4 {\mathcal H_{wm}=\left\{x\in\mathcal H\,;\,\lim_{N\rightarrow+\infty}\dfrac{1}{N}\displaystyle\sum_{n=1}^N\left|(T^nx,y)\right|=0,\ \forall\,y\in\mathcal H\right\}}.

Proof: By Lemma 4 of ERT8,

\displaystyle \dfrac{1}{N}\sum_{n=1}^N\left|(T^nx,y)\right|\rightarrow 0\iff \dfrac{1}{N}\sum_{n=1}^N\left|(T^nx,y)\right|^2\rightarrow 0.

This last expression may be rewritten as

\displaystyle \dfrac{1}{N}\sum_{n=1}^N (T^nx,y)(y,T^nx)=\left(\dfrac{1}{N}\sum_{n=1}^N(y,T^nx)T^nx,y\right),

so it is enough to prove that

\displaystyle \dfrac{1}{N}\sum_{n=1}^N(y,T^nx)T^nx\rightarrow 0\,.

We use van der Corput trick to this matter: define {x_n=(y,T^nx)T^nx}. Then

\displaystyle \begin{array}{rcl} (x_{n+h},x_n)&=&\left((y,T^{n+h}x)T^{n+h}x,(y,T^nx)T^nx\right)\\ &=&(y,T^{n+h}x)\cdot(T^nx,y)\cdot(T^{n+h}x,T^nx)\\ &=&(y,T^{n+h}x)\cdot(T^nx,y)\cdot(T^{h}x,x)\,, \end{array}

whose absolute value is bounded by {\|x\|^2\cdot\|y\|^2\cdot|(T^hx,x)|} and so

\displaystyle \dfrac{1}{H}\sum_{h=1}^H\dfrac{1}{N}\sum_{n=1}^N (x_{n+h},x_n)\le\|x\|^2\cdot\|y\|^2\cdot\dfrac{1}{H}\sum_{h=1}^H|(T^hx,x)|

which goes to zero by assumption. This proves (1) and concludes the proof. \Box

We now proceed to the announced result.

Theorem 5 (Koopman-von Neumann) {\mathcal H=\mathcal H_c\oplus^\perp\mathcal H_{wm}}.

Proof: First, note that if {x\in\mathcal H_{wm}} and {y\in\mathcal H_c}, say {Ty=\lambda y}, then

\displaystyle \begin{array}{rcl} \dfrac{1}{N}\sum_{n=1}^N |(T^nx,y)|&=&\dfrac{1}{N}\sum_{n=1}^N|(x,T^{-n}y)|\\ &&\\ &=&\dfrac{1}{N}\sum_{n=1}^N|(x,y)|\\&&\\ &=&|(x,y)| \end{array}

and so {(x,y)=0}. This proves the inclusion {\mathcal H_{wm}\subset{\mathcal H_c}^\perp}.

For the reverse inclusion, let {x\in{\mathcal H_c}^\perp}. We want to show that {x} is weak mixing, that is, that

\displaystyle \dfrac{1}{N}\sum_{n=1}^N |(T^nx,x)|\rightarrow 0.

Consider, for each {N\in\mathbb N}, the Hilbert-Schmidt operator (see the Appendix) {S_N:\mathcal H\rightarrow\mathcal H} defined as

\displaystyle S_Ny=\dfrac{1}{N}\sum_{n=1}^N (y,T^nx)T^nx

and the operator {Sy=\lim S_Ny}. The claim will follow if we manage to prove that {(x,Sx)=0}. We have

1. {S} is a well-defined linear operator: what we can guarantee is that, for every {z\in\mathcal H}, {\lim(z,S_Ny)} exists. In fact, {T\times T} is a unitary linear operator on the tensor product {\mathcal H\otimes\mathcal H}, whose inner-product is

{(y_1\otimes z_1,y_2\otimes z_2)_{\mathcal H\otimes\mathcal H}=(y_1,z_1)_{\mathcal H}\cdot(y_2,z_2)_{\mathcal H}},

and so, by von Neumann theorem, the limit of

\displaystyle \begin{array}{rcl} (z,S_Ny)&=&\sum_{n=1}^N(y,T^nx)(z,T^nx)\\ &=&\sum_{n=1}^N (y\otimes z,(T\times T)^n(x\otimes x)) \end{array}

exists. By Riesz representation theorem, there exists a unique {Sy\in\mathcal H} for which

\displaystyle (z,Sy)=\lim_{N\rightarrow+\infty}(z,S_Ny)\,,\ \forall\, z\in\mathcal H.

The linearity of {S} follows from the linearity of the inner-product.

2. {S} is a compact operator: we remark that, being defined as the weak limit of {S_N}, {S} is not compact by definition. This is the reason we consider the Hilbert-Schmidt operators.

For each {N}, the norm of {S_N} is at most {\|x\|^2}. Then {S} is the weak limit of a sequence of bounded Hilbert-Schmidt operator. By the Appendix, {S} is compact.

3. {S} commutes with {T}: this follows from the “almost” commutativity of {S_N} and {T}. In fact,

\displaystyle \begin{array}{rcl} S_NTy-TS_Ny&=&\dfrac{1}{N}\sum_{n=1}^N (Ty,T^nx)T^nx-\dfrac{1}{N}\sum_{n=1}^N (y,T^nx)T^{n+1}x\\&&\\ &=&\dfrac{1}{N}\sum_{n=0}^{N-1}(y,T^nx)T^{n+1}x-\dfrac{1}{N}\sum_{n=1}^N (y,T^nx)T^{n+1}x\\ &&\\&=&T\left(\dfrac{(y,x)x-(y,T^Nx)T^Nx}{N}\right) \end{array}

and this last expression goes to zero as {N\rightarrow+\infty}.

4. For any {y\in\mathcal H}, {Sy} is almost periodic: in fact,

\displaystyle \{T^nSy\,;\,n\in\mathbb N\}=\{ST^nf\,;\,n\in\mathbb N\}

is pre-compact, by the compactness of {S}.

By hypothesis, {x} is orthogonal to every almost periodic element. In particular, {(x,Sx)=0}, completing the proof. \Box

Going back to the ergodic theoretical setup, we obtain

Corollary 6 The mps {\mathbb X} is weak mixing iff {\mathcal H_c=\emptyset} and compact iff {\mathcal H_{wm}=\emptyset}.

Let {\mathcal Z_1} be the {\sigma}-algebra generated by the eigenfunctions of {T} (it is the smallest {\sigma}-algebra for which every eigenfunction is measurable).

Definition 7 {\mathcal Z_1} is called the Kronecker factor of {\mathbb X}.

One can show that {\mathcal Z_1} is the inverse limit of a sequence of {\sigma}-algebras, each of them being isomorphic to a rotation on a abelian group. Such explicit characterization allows to prove the existence of double ergodic averages, but this is a topic for later posts.

1. Appendix: Hilbert-Schmidt operators

Definition 8 A continuous linear operator {T:\mathcal H\rightarrow\mathcal H} is Hilbert-Schmidt if there exists an orthonormal basis {\{x_i\}_{i\in I}} such that

\displaystyle {\|T\|_{\rm{HS}}}^2=\sum_{i\in I}\|Tx_i\|^2<+\infty.

In this case, the norm of {T} is defined as {\|T\|_{\rm{HS}}}. The main properties we used above are:

HS1. Every Hilbert-Schmidt operator is compact.

HS2. The limit, in either the norm, strong or weak norm topologies, of a sequence of bounded Hilbert-Schmidt operators is Hilbert-Schmidt.

The interested reader is invited to read this post of Terence Tao.

Previous posts: ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7, ERT8, ERT9, ERT10, ERT11.


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