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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 815726, 5 pages
Linear Sequences and Weighted Ergodic Theorems
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Received 14 February 2013; Accepted 23 April 2013
Academic Editor: Baodong Zheng
Copyright © 2013 Tanja Eisner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form for a measure preserving system and , avoiding in the latter case the problem of finding the full measure set of appropriate points .
The classical mean and pointwise ergodic theorems due to von Neumann and Birkhoff, respectively, take their origin in questions from statistical physics and found applications in quite different areas of mathematics such as number theory, stochastics, and harmonic analysis. Over the years, they were extended and generalised in many ways. For example, to multiple ergodic theorems, see Furstenberg , Bergelson et al. , Host and Kra , Ziegler , and Tao , to the Wiener-Wintner theorem, see Assani , Lesigne , Frantzikinakis , Host and Kra , and Eisner and Zorin-Kranich , to the return time theorem and its generalisations, see Bourgain et al. , Demeter et al. , Rudolph , Assani and Presser [14, 15], and Zorin-Kranich , and to further weighted, modulated, and subsequential ergodic theorems, see Berend et al. , Below and Losert , Bourgain [19, 20], and Wierdl .
The return time theorem due to Bourgain, solving a quite long standing open problem, is a classical example of a weighted pointwise ergodic theorem. It states that for every measure preserving system and , the sequence is for -almost every a good weight for the pointwise ergodic theorem. This means that for every other system and every , the weighted ergodic averages converge almost everywhere in . The proof due to Bourgain et al. , see also Lesigne et al.  and Zorin-Kranich , is descriptive and gives conditions on to produce a good weight. However, these conditions can be quite difficult to check in a concrete situation. Later, Rudolph , see also Assani and Presser  and Zorin-Kranich , gave a generalisation of the return time theorem and showed that (in the previous notation) the sequence is for almost every a universally good weight for multiple ergodic averages; see Definition 4 later. However, the conditions on the point did not become easier to check.
The most general class of systems for which the convergence in the multiple return time theorem is known to hold everywhere, hence, leading to good weights which are easy to construct, are nilsystems, that is, systems of the form for a nilpotent Lie group , a discrete cocompact subgroup , the Haar measure on , and the rotation by some element of . For such system , and , the sequence is called a basic nilsequence. A nilsequence is a uniform limit of basic nilsequences of the same step, or, equivalently, a sequence of the form for an inverse limit of nilsystems of the same step, , a rotation on and ; see Host and Maass . Indeed, recently Zorin-Kranich  proved the Wiener-Wintner type return time theorem for nilsequences showing universal convergence of averages for every and every nilsequence , where the universal sets of convergence do not depend on . This generalised an earlier result by Assani et al.  for sequences of the form , , and .
In this paper, we search for good weights for ergodic theorems using a functional analytic perspective and produce deterministic good weights. We first introduce sequences of the form , which we call linear sequences if is in a Banach space , and is a linear operator on with relatively weakly compact orbits; see Section 2 later. Using a structure result for linear sequences, we show that they are good weights for the multiple polynomial ergodic theorem (Section 4) and for the Wiener-Wintner type multiple return time theorem discussed (Section 3). In the last section, we present a counterexample showing that the assumption on the operators cannot be dropped even for positive isometries on Banach lattices and the mean ergodic theorem.
We finally remark that all results in this paper hold if we replace linear sequences by a larger class of “asymptotic nilsequences,” that is, for sequences of the form , where is a nilsequence and is a bounded sequence satisfying (cf. Theorem 3). Examples of asymptotic nilsequences (of step ≥2 in general) are multiple polynomial correlation sequences of the form for an ergodic invertible measure preserving system , , , and polynomials with integer coefficients, . This follows from Leibman [26, Theorem 3.1] and, in the case of linear polynomials, is due to Bergelson et al. [27, Theorem 1.9]. Thus, multiple polynomial correlation sequences provide another class of deterministic examples of good weights for the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem discussed in Sections 3 and 4.
2. Linear Sequences and Their Structure
A linear operator on a Banach space has relatively weakly compact orbits if for every , the orbit is relatively weakly compact in .
Definition 1. We call a sequence a linear sequence if there exists an operator on a Banach space with relatively weakly compact orbits and , , such that holds for every .
A large class of operators with relatively weakly compact orbits, leading to a large class of linear sequences, are power bounded operators on reflexive Banach spaces. Recall that an operator is called power bounded if it satisfies . Another class of operators with relatively weakly compact orbits are power bounded positive operators on a Banach lattice preserving the order interval generated by a strictly positive function; see, for example, Schaefer [28, Theorem II.5.10(f) and Proposition II.8.3]. See [29, Section I.1] and [30, Section 16.1] for further discussion.
Remark 2. By restricting to the closed linear invariant subspace induced by the orbit and using the decomposition for , it suffices to assume that only the relevant orbit is relatively weakly compact in the definition of a linear sequence . Note that in this case has relatively weakly compact orbits on by a limiting argument; see, for example, [29, Lemma I.1.6].
We obtain the following structure result for linear sequences as a direct consequence of an extended Jacobs-Glicksberg-deLeeuw decomposition for operators with relatively weakly compact orbits.
Theorem 3. Every linear sequence is a sum of an almost periodic sequence and a (bounded) sequence satisfying .
Proof. Let be an operator on a Banach space with relatively weakly compact orbits. By the Jacobs-Glicksberg-deLeeuw decomposition, see, for example, [29, Theorem II.4.8], , where
while every satisfies for every . (Recall that by the Koopman-von Neumann lemma, see, for example, Petersen [31, p. 65], for bounded sequences the condition is equivalent to for some subsequence with density .)
Let , and define the sequence by . For we have by the aforementioned. If now is an eigenvector corresponding to an eigenvalue , then . Therefore, for every , the sequence is a uniform limit of finite linear combinations of sequences , , and is therefore almost periodic. The assertion follows.
3. A Wiener-Wintner Type Result for the Multiple Return Time Theorem
In this section, we show that one can take linear sequences as weights in the multiple Wiener-Wintner type generalisation of the return time theorem due to Zorin-Kranich  and Assani et al.  discussed in the introduction.
First we recall the definition of a property satisfied universally.
Definition 4. Let and be a pointwise property for measure preserving dynamical systems. We say that a property is satisfied universally almost everywhere if for every system and every there is a set of full measure such that for every and every system … for every system and there is a set of full measure such that for every the property holds.
We show the following linear version of the Wiener-Wintner type multiple return time theorem.
Theorem 5. For every , the weighted averages (2) converge universally almost everywhere for every linear sequence , where the universal sets , , of full measure are independent of .
Proof. By Theorem 3, we can show the assertion for almost periodic sequences and for satisfying separately. For sequences from the second class, the assertion follows from the estimate
with a clear choice of .
Universal convergence for almost periodic sequences is a consequence of Zorin-Kranich’s result [16, Theorem 1.3] which shows the assertion for the larger class of nilsequences.
4. Weighted Multiple Polynomial Ergodic Theorem
Using the Host-Kra Wiener-Wintner type result for nilsequences and extending their result for linear polynomials from , Chu  showed the following (see also  for a slightly different proof). Let be a system and . Then, for almost every , the sequence is a good weight for the multiple polynomial ergodic theorem; that is, for the sequence of weights given by and for every , the weighted multiple polynomial averages converge in for every system with invertible , every , and every polynomial with integer coefficients.
The following result is a consequence of Chu [32, Theorem 1.3], with the fact that the product of two nilsequences is again a nilsequence and equidistribution theory for nilsystems; see, for example, Parry  and Leibman .
Theorem 6. Every nilsequence is a good weight for the multiple polynomial ergodic theorem.
This remains true when replacing a nilsequence by a linear sequence.
Theorem 7. Every linear sequence is a good weight for the multiple polynomial ergodic theorem.
Proof. For an almost periodic sequence , the averages (6) converge in by Theorem 6. It is also clear that the averages (6) converge to in for every sequence satisfying . The assertion follows now from Theorem 3.
5. A Counter Example
The following example shows that if one does not assume relative weak compactness in the definition of linear sequences, each of the previous results can fail dramatically even for positive isometries on Banach lattices.
Example 8. Let and be the right shift operator; that is,
We first show that for every , , and , we have
Indeed, take and such that . Then, for we have
Choosing, for example, finishes the proof of (8).
In particular, for , we see that the sequence is Cesàro divergent for every with and for every which is Cesàro divergent. Note that the sets of such and are open and dense in and , respectively. (The assertion for is clear as well as the openness of the set of Cesàro divergent sequences in , and density follows from the fact that one can construct Cèsaro divergent sequences of arbitrarily small supremum norm.) Thus, for topologically very big sets of and (with complements being nowhere dense), the sequence is not a good weight for the mean ergodic theorem.
We further show that in fact for every , there is so that for every from a dense open set, the sequence is Cesàro divergent, implying that the sequence is not a good weight for the mean ergodic theorem.
Take and define the function on the unit disc by . Then, is a nonzero holomorphic function belonging to the Hardy space . By Hardy space theory, see, for example, Rosenblum and Rovnyak [35, Theorem 4.25], there is a set of positive Lebesgue measure such that for every , we have For every such , by (8), we see that the sequence is Cesàro divergent for every such that is Cesàro divergent. The set of such is open and dense in since it is the case for , and the multiplication operator is an invertible isometry. Thus, for every , there is an open dense set of such that the sequence fails to be a good weight for the mean ergodic theorem.
The author thanks Pavel Zorin-Kranich for helpful discussions.
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