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 -algebra 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 of Furstenberg theorem: this is known as Roth theorem, who first proved the existence of arithmetic progressions of length 3 in subsets of 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 . Here, stands for the -action defined by the mps and also for the Koopman-von Neumann operator . More specifically, is weak mixing iff
for any such that , and compact iff is pre-compact in for any .
Yet considering the spectral point of view, let be a Hilbert space and a unitary operator. Motivated by the above situation, let us make the following
Definition 1 An element is weak mixing if
and almost periodic if the set is pre-compact in .
Consider the sets
The main theorem of this post, due to Koopman and von Neumann, estabilishes the equality . In order to do this, we characterize and in different manners.
Proposition 2 is equal to the closure of the union of eigenspaces:
Proof: The inclusion is clear: if , the closure of is equal to , where is a subgroup of . To the reverse inclusion, consider almost periodic and let . The restriction map is a compact operator. Being also self-adjoint, the spectral theorem implies the existence of an orthonormal basis of eigenvectors of . In particular, is in the closure of the subspace generated by them, which concludes the proof.
The alternative characterization of uses the van der Corput trick (see ERT9). Let us remember it.
Theorem 3 (van der Corput trick) If is a bounded sequence in a Hilbert space and if
Proposition 4 .
Proof: By Lemma 4 of ERT8,
This last expression may be rewritten as
so it is enough to prove that
We use van der Corput trick to this matter: define . Then
whose absolute value is bounded by and so
which goes to zero by assumption. This proves (1) and concludes the proof.
We now proceed to the announced result.
Theorem 5 (Koopman-von Neumann) .
Proof: First, note that if and , say , then
and so . This proves the inclusion .
For the reverse inclusion, let . We want to show that is weak mixing, that is, that
Consider, for each , the Hilbert-Schmidt operator (see the Appendix) defined as
and the operator . The claim will follow if we manage to prove that . We have
1. is a well-defined linear operator: what we can guarantee is that, for every , exists. In fact, is a unitary linear operator on the tensor product , whose inner-product is
and so, by von Neumann theorem, the limit of
exists. By Riesz representation theorem, there exists a unique for which
The linearity of follows from the linearity of the inner-product.
2. is a compact operator: we remark that, being defined as the weak limit of , is not compact by definition. This is the reason we consider the Hilbert-Schmidt operators.
For each , the norm of is at most . Then is the weak limit of a sequence of bounded Hilbert-Schmidt operator. By the Appendix, is compact.
3. commutes with : this follows from the “almost” commutativity of and . In fact,
and this last expression goes to zero as .
4. For any , is almost periodic: in fact,
is pre-compact, by the compactness of .
By hypothesis, is orthogonal to every almost periodic element. In particular, , completing the proof.
Going back to the ergodic theoretical setup, we obtain
Corollary 6 The mps is weak mixing iff and compact iff .
Let be the -algebra generated by the eigenfunctions of (it is the smallest -algebra for which every eigenfunction is measurable).
Definition 7 is called the Kronecker factor of .
One can show that is the inverse limit of a sequence of -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 is Hilbert-Schmidt if there exists an orthonormal basis such that
In this case, the norm of is defined as . 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.