research article linear sequences and weighted ergodic theorems · 2019. 7. 31. · 4. weighted...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 815726 5 pageshttpdxdoiorg1011552013815726

Research ArticleLinear Sequences and Weighted Ergodic Theorems

Tanja Eisner

Korteweg-de Vries Institute for Mathematics University of Amsterdam PO Box 94248 1090 GE Amsterdam The Netherlands

Correspondence should be addressed to Tanja Eisner talofauni-tuebingende

Received 14 February 2013 Accepted 23 April 2013

Academic Editor Baodong Zheng

Copyright copy 2013 Tanja Eisner This is an open access article distributed under the Creative Commons Attribution License whichpermits 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 theWiener-Wintner type multiplereturn time theorem and themultiple polynomial ergodic theoremThese weights are deterministic and come from orbits of certainbounded linear operators on Banach spacesThis extends the known results for nilsequences and return time sequences of the form(g(S119899y)) for a measure preserving system (Y S) and 119892 isin 119871

infin

(119884) avoiding in the latter case the problem of finding the full measureset of appropriate points y

1 Introduction

The classical mean and pointwise ergodic theorems due tovon Neumann and Birkhoff respectively take their origin inquestions from statistical physics and found applications inquite different areas of mathematics such as number theorystochastics and harmonic analysis Over the years they wereextended and generalised inmanyways For example tomul-tiple ergodic theorems see Furstenberg [1] Bergelson et al[2] Host and Kra [3] Ziegler [4] and Tao [5] to theWiener-Wintner theorem see Assani [6] Lesigne [7] Frantzikinakis[8] Host and Kra [9] and Eisner and Zorin-Kranich [10] tothe return time theorem and its generalisations see Bourgainet al [11] Demeter et al [12] Rudolph [13] Assani and Presser[14 15] and Zorin-Kranich [16] and to further weightedmodulated and subsequential ergodic theorems see Berendet al [17] Below and Losert [18] Bourgain [19 20] andWierdl [21]

The return time theorem due to Bourgain solving aquite long standing open problem is a classical example ofa weighted pointwise ergodic theorem It states that for everymeasure preserving system (119884 120583 119878) and 119892 isin 119871

infin

(119884 120583) thesequence (119892(119878

119899

119910)) is for 120583-almost every 119910 a good weight forthe pointwise ergodic theorem This means that for everyother system (119884

1 1205831 1198781) and every 119892

1isin 119871infin

(1198841 1205831) the

weighted ergodic averages

1

119873

119873

sum

119899=1

119892 (119878119899

119910) 1198921(119878119899

11199101) (1)

converge almost everywhere in1199101The proof due to Bourgain

et al [11] see also Lesigne et al [22] and Zorin-Kranich [23]is descriptive and gives conditions on 119910 to produce a goodweight However these conditions can be quite difficult tocheck in a concrete situation Later Rudolph [13] see alsoAssani and Presser [14] and Zorin-Kranich [16] gave a gener-alisation of the return time theorem and showed that (in theprevious notation) the sequence (119892(119878

119899

119910)) is for almost every119910 a universally good weight for multiple ergodic averages seeDefinition 4 later However the conditions on the point 119910 didnot become easier to check

The most general class of systems for which the conver-gence in the multiple return time theorem is known to holdeverywhere hence leading to good weights which are easyto construct are nilsystems that is systems of the form119884 = 119866Γ for a nilpotent Lie group 119866 a discrete cocompactsubgroup Γ the Haarmeasure 120583 on119866Γ and the rotation 119878 bysome element of 119866 For such system (119884 120583 119878) 119892 isin 119862(119884) and119910 isin 119884 the sequence (119892(119878

119899

119910)) is called a basic nilsequenceA nilsequence is a uniform limit of basic nilsequences of thesame step or equivalently a sequence of the form (119892(119878

119899

119910))

for an inverse limit 119884 of nilsystems of the same step 119910 isin 119884a rotation 119878 on 119884 and 119892 isin 119862(119884) see Host and Maass[24] Indeed recently Zorin-Kranich [16] proved theWiener-Wintner type return time theorem for nilsequences showinguniversal convergence of averages

1

119873

119873

sum

119899=1

1198861198991198921(119878119899

11199101) sdot sdot sdot 119892

119896(119878119899

119896119910119896) (2)

2 Abstract and Applied Analysis

for every 119896 isin N and every nilsequence (119886119899) where the uni-

versal sets of convergence do not depend on (119886119899) This gener-

alised an earlier result byAssani et al [25] for sequences of theform (120582

119899

) 120582 isin T and 119896 = 2In this paper we search for good weights for ergodic

theorems using a functional analytic perspective and producedeterministic good weights We first introduce sequences ofthe form (⟨119879

119899

119909 1199091015840

⟩) which we call linear sequences if 119909 is in aBanach space119883 1199091015840 isin 119883

1015840 and 119879 is a linear operator on119883withrelatively weakly compact orbits see Section 2 later Using astructure result for linear sequences we show that they aregood weights for the multiple polynomial ergodic theorem(Section 4) and for the Wiener-Wintner type multiple returntime theorem discussed (Section 3) In the last section wepresent a counterexample showing that the assumption on theoperators cannot be dropped even for positive isometries onBanach lattices and the mean ergodic theorem

We finally remark that all results in this paper hold if wereplace linear sequences by a larger class of ldquoasymptotic nilse-quencesrdquo that is for sequences (119886

119899) of the form 119886

119899= 119887119899+ 119888119899

where (119887119899) is a nilsequence and (119888

119899) is a bounded sequence

satisfying lim119873rarrinfin

(1119873)sum119873

119899=1|119888119899| = 0 (cf Theorem 3)

Examples of asymptotic nilsequences (of step ge2 in general)aremultiple polynomial correlation sequences (119886

119899) of the form

119886119899= int119884

1198781199011(119899)

1198921sdot sdot sdot 119878119901119896(119899)

119892119896119889120583 (3)

for an ergodic invertible measure preserving system (119884 120583 119878)119896 isin N 119892

119895isin 119871infin

(119884 120583) and polynomials 119901119895with integer

coefficients 119895 = 1 119896 This follows from Leibman [26Theorem 31] and in the case of linear polynomials is due toBergelson et al [27 Theorem 19] Thus multiple polynomialcorrelation sequences provide another class of deterministicexamples of good weights for the Wiener-Wintner typemultiple return time theorem and the multiple polynomialergodic theorem discussed in Sections 3 and 4

2 Linear Sequences and Their Structure

A linear operator 119879 on a Banach space119883 has relatively weaklycompact orbits if for every 119909 isin 119883 the orbit 119879119899119909 119899 isin N

0 is

relatively weakly compact in119883

Definition 1 We call a sequence (119886119899) sub C a linear sequence if

there exists an operator119879 on a Banach space119883with relativelyweakly compact orbits and 119909 isin 119883 1199091015840 isin 119883

1015840 such that 119886119899=

⟨119879119899

119909 1199091015840

⟩ holds for every 119899 isin N

A large class of operators with relatively weakly compactorbits leading to a large class of linear sequences are powerbounded operators on reflexive Banach spaces Recall that anoperator119879 is called power bounded if it satisfies sup

119899isinN119879119899

lt

infin Another class of operators with relatively weakly compactorbits are power bounded positive operators on a Banachlattice 119871

1

(120583) preserving the order interval generated by astrictly positive function see for example Schaefer [28The-orem II510(f) and Proposition II83] See [29 Section I1]and [30 Section 161] for further discussion

Remark 2 By restricting to the closed linear invariant sub-space 119884 = lin119879119899119909 119899 isin N

0 induced by the orbit and using

the decomposition 1198831015840

= 1198841015840

oplus 1198841015840

0for 11988410158400= 1199091015840

1199091015840

|119884= 0 it

suffices to assume that only the relevant orbit 119879119899119909 119899 isin N0 is

relatively weakly compact in the definition of a linearsequence (⟨119879

119899

119909 1199091015840

⟩) Note that in this case 119879 has relativelyweakly compact orbits on 119884 by a limiting argument see forexample [29 Lemma I16]

Weobtain the following structure result for linear sequen-ces as a direct consequence of an extended Jacobs-Glicksberg-deLeeuw decomposition for operators with relatively weaklycompact orbits

Theorem 3 Every linear sequence is a sum of an almostperiodic sequence and a (bounded) sequence (119888

119899) satisfying

lim119873rarrinfin

(1119873)sum119873

119899=1|119888119899| = 0

Proof Let 119879 be an operator on a Banach space 119883 withrelatively weakly compact orbits By the Jacobs-Glicksberg-deLeeuw decomposition see for example [29 TheoremII48]119883 = 119883

119903oplus 119883119904 where

119883119903= lin 119909 119879119909 = 120582119909 for some 120582 isin T (4)

while every 119909 isin 119883119904

satisfieslim119873rarrinfin

(1119873)sum119873

119899=1|⟨119879119899

119909 1199091015840

⟩| = 0 for every 1199091015840

isin 1198831015840

(Recall that by the Koopman-von Neumann lemma seefor example Petersen [31 p 65] for bounded sequences thecondition lim

119873rarrinfin(1119873)sum

119873

119899=1|119888119899| = 0 is equivalent to

lim119895rarrinfin

119888119899119895

= 0 for some subsequence 119899119895 sub N with density

1)Let 119909 isin 119883 1199091015840 isin 119883

1015840 and define the sequence (119886119899) by 119886119899=

⟨119879119899

119909 1199091015840

⟩ For 119909 isin 119883119904we have lim


119873

119899=1|119886119899| = 0

by the aforementioned If now 119909 is an eigenvector corre-sponding to an eigenvalue 120582 isin T then 119886

119899= 120582119899

⟨119909 1199091015840

⟩Therefore for every 119909 isin 119883

119903 the sequence (119886

119899) is a uniform

limit of finite linear combinations of sequences (120582119899) 120582 isin T and is therefore almost periodic The assertion follows

3 A Wiener-Wintner Type Result forthe Multiple Return Time Theorem

In this section we show that one can take linear sequences asweights in the multiple Wiener-Wintner type generalisationof the return time theorem due to Zorin-Kranich [16] andAssani et al [25] discussed in the introduction

First we recall the definition of a property satisfieduniversally

Definition 4 Let 119896 isin N and 119875 be a pointwise property for 119896measure preserving dynamical systems We say that a prop-erty 119875 is satisfied universally almost everywhere if for everysystem (119884

1 1205831 1198781) and every 119892

1isin 119871infin

(1198841 1205831) there is a set

1198841015840

1sub 1198841of full measure such that for every 119910

1isin 1198841015840

1and

every system (1198842 1205832 1198782) for every system (119884

119896 120583119896 119878119896) and

119892119896isin 119871infin

(119884119896 120583119896) there is a set 1198841015840

119896sub 119884119896of full measure such

that for every 119910119896isin 1198841015840

119896the property 119875 holds

Abstract and Applied Analysis 3

We show the following linear version of theWiener-Wint-ner type multiple return time theorem

Theorem 5 For every 119896 isin N the weighted averages (2) con-verge universally almost everywhere for every linear sequence(119886119899) where the universal sets 1198841015840

119895 119895 = 1 119896 of full measure

are independent of (119886119899)

Proof By Theorem 3 we can show the assertion foralmost periodic sequences and for (119886

119899) satisfying

lim119873rarrinfin

(1119873)sum119873

119899=1|119886119899| = 0 separately For sequences

from the second class the assertion follows from the estimate1003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

1198861198991198921(119878119899

11199101) sdot sdot sdot 119892

119896(119878119899

119896119910119896)

1003816100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198921

1003817100381710038171003817infin sdot sdot sdot1003817100381710038171003817119892119896

1003817100381710038171003817infin

1

119873

119873

sum

119899=1

10038161003816100381610038161198861198991003816100381610038161003816

(5)

with a clear choice of 11988410158401 119884

1015840

119896

Universal convergence for almost periodic sequences isa consequence of Zorin-Kranichrsquos result [16 Theorem 13]which shows the assertion for the larger class of nilsequences

4 Weighted Multiple PolynomialErgodic Theorem

Using the Host-Kra Wiener-Wintner type result for nilse-quences and extending their result for linear polynomialsfrom [9] Chu [32] showed the following (see also [10] fora slightly different proof) Let (119884 120583 119878) be a system and 119892 isin

119871infin

(119884 120583) Then for almost every 119910 isin 119884 the sequence(119892(119878119899

119910)) is a good weight for the multiple polynomial ergodictheorem that is for the sequence of weights (119886

119899) given by

119886119899

= 119892(119878119899

119910) and for every 119896 isin N the weighted multiplepolynomial averages

1

119873

119873

sum

119899=1

1198861198991198781199011(119899)

11198921sdot sdot sdot 119878119901119896(119899)

1119892119896

(6)

converge in 1198712 for every system (119884

1 1205831 1198781) with invertible

1198781 every 119892

1 119892


(1198841 1205831) and every polynomial

1199011 119901

119896with integer coefficients

The following result is a consequence of Chu [32 Theo-rem 13] with the fact that the product of two nilsequencesis again a nilsequence and equidistribution theory for nilsys-tems see for example Parry [33] and Leibman [34]

Theorem6 Every nilsequence is a goodweight for themultiplepolynomial ergodic theorem

This remains true when replacing a nilsequence by a lin-ear sequence

Theorem7 Every linear sequence is a goodweight for themul-tiple polynomial ergodic theorem

Proof For an almost periodic sequence (119886119899) the averages (6)

converge in 1198712 byTheorem 6 It is also clear that the averages

(6) converge to 0 in 119871infin for every sequence (119886

119899) satisfying

lim119873rarrinfin

(1119873)sum119873

119899=1|119886119899| = 0The assertion follows now from

Theorem 3

5 A Counter Example

The following example shows that if one does not assume rel-ative weak compactness in the definition of linear sequenceseach of the previous results can fail dramatically even forpositive isometries on Banach lattices

Example 8 Let119883 = 1198971 and 119879 be the right shift operator that

is

119879 (1199051 1199052 ) = (0 119905

1 1199052 ) (7)

We first show that for every 120582 isin T 119909 = (119905119895) isin 119883 and 119909

1015840

=

(119904119895) isin 1198831015840 we have

lim119873rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

⟨119879119899

119909 1199091015840

⟩ minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (8)

Indeed take 120576 gt 0 and 119869 isin N such that suminfin119895=119869+1

|119905119895| lt 120576 Then

for119873 isin N we have10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

⟨119879119899

119909 1199091015840

⟩ minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

infin

sum

119895=1

119905119895119904119899+119895

minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

119869

sum

119895=1

119905119895119904119899+119895

minus1

119873

119873

sum

119899=1

120582119899

119904119899

119869

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ 210038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119869

sum

119895=1

120582119895

119905119895

1

119873

119873+119895

sum

119899=1+119895

120582119899

119904119899minus

119869

sum

119895=1

120582119895

119905119895

1

119873

119873

sum

119899=1

120582119899

119904119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ 210038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

le

21198691199091

10038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

119873+ 2

10038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

(9)

Choosing for example119873 gt 1198691199091120576 finishes the proof of (8)

In particular for 120582 = 1 we see that the sequence(⟨119879119899

119909 1199091015840

⟩) is Cesaro divergent for every 119909 = (119905119895) isin 1198971 with

suminfin

119895=1119905119895

= 0 and for every 1199091015840

isin 119897infin which is Cesaro divergent

Note that the sets of such 119909 and 1199091015840 are open and dense in 119897

1

and 119897infin respectively (The assertion for 1198971 is clear as well as the

openness of the set of Cesaro divergent sequences in 119897infin and

density follows from the fact that one can construct Cesarodivergent sequences of arbitrarily small supremum norm)Thus for topologically very big sets of 119909 and 119909

1015840 (with com-plements being nowhere dense) the sequence (⟨119879

119899

119909 1199091015840

⟩) isnot a good weight for the mean ergodic theorem

We further show that in fact for every 0 = 119909 isin 1198971 there

is 120582 isin T so that for every 1199091015840

isin 119897infin from a dense open set

the sequence (120582119899⟨119879119899119909 1199091015840⟩) is Cesaro divergent implying thatthe sequence (⟨119879

119899

119909 1199091015840

⟩) is not a good weight for the meanergodic theorem


Take 0 = 119909 = (119905119895) isin 1198971 and define the function 119891 on the

unit disc D by 119891(119911) = suminfin

119895=1119905119895119911119895 Then 119891 is a nonzero holo-

morphic function belonging to the Hardy space 1198671

(D) ByHardy space theory see for example Rosenblum andRovnyak [35 Theorem 425] there is a set 119872 sub T of positiveLebesgue measure such that for every 120582 isin 119872 we have

lim119903rarr1minus

119891 (119903120582) =

infin

sum

119895=1

120582119895

119905119895

= 0 (10)

For every such 120582 by (8) we see that the sequence(120582119899

⟨119879119899

119909 1199091015840

⟩) is Cesaro divergent for every 1199091015840

= (119904119895) isin 119897infin

such that (120582119895119904119895) is Cesaro divergentThe set of such 119909

1015840 is openand dense in 119897

infin since it is the case for 120582 = 1 and the mul-tiplication operator (119904

119895) 997891rarr (120582

119895

119904119895) is an invertible isometry

Thus for every 0 = 119909 isin 1198971 there is an open dense set of1199091015840 isin 119897

infin

such that the sequence (⟨119879119899119909 1199091015840⟩) fails to be a goodweight forthe mean ergodic theorem

Acknowledgment

The author thanks Pavel Zorin-Kranich for helpful discus-sions

References

[1] H Furstenberg Recurrence in Ergodic Theory and Combinato-rial Number Theory Princeton University Press Princeton NJUSA 1981

[2] V Bergelson A Leibman and E Lesigne ldquoIntersective poly-nomials and the polynomial Szemeredi theoremrdquo Advances inMathematics vol 219 no 1 pp 369ndash388 2008

[3] B Host and B Kra ldquoNonconventional ergodic averages andnilmanifoldsrdquo Annals of Mathematics vol 161 no 1 pp 397ndash488 2005

[4] T Ziegler ldquoUniversal characteristic factors and Furstenbergaveragesrdquo Journal of the AmericanMathematical Society vol 20no 1 pp 53ndash97 2007

[5] T Tao ldquoNorm convergence of multiple ergodic averages forcommuting transformationsrdquo Ergodic Theory and DynamicalSystems vol 28 no 2 pp 657ndash688 2008

[6] I Assani Wiener Wintner Ergodic Theorems World ScientificPublishing River Edge NJ USA 2003

[7] E Lesigne ldquoSpectre quasi-discret et theoreme ergodique deWiener-Wintner pour les polynomesrdquo Ergodic Theory andDynamical Systems vol 13 no 4 pp 767ndash784 1993

[8] N Frantzikinakis ldquoUniformity in the polynomialWiener-Wint-ner theoremrdquo Ergodic Theory and Dynamical Systems vol 26no 4 pp 1061ndash1071 2006

[9] B Host and B Kra ldquoUniformity seminorms on ℓinfin and appli-

cationsrdquo Journal drsquoAnalyse Mathematique vol 108 pp 219ndash2762009

[10] T Eisner and P Zorin-Kranich ldquoUniformity in the Wiener-Wintner theorem for nilsequencesrdquo Discrete and ContinuousDynamical Systems A vol 33 no 8 pp 3497ndash3516 2013

[11] J Bourgain H Furstenberg Y Katznelson and D S OrnsteinldquoAppendix on return-time sequencesrdquo Publications Mathemati-ques de lrsquoInstitut des Hautes Etudes Scientifiques vol 69 pp 42ndash45 1989

[12] C Demeter M T Lacey T Tao and C Thiele ldquoBreaking theduality in the return times theoremrdquo Duke Mathematical Jour-nal vol 143 no 2 pp 281ndash355 2008

[13] D J Rudolph ldquoFully generic sequences and a multiple-termreturn-times theoremrdquo InventionesMathematicae vol 131 no 1pp 199ndash228 1998

[14] I Assani and K Presser ldquoPointwise characteristic factors for themultiterm return times theoremrdquo Ergodic Theory and Dynami-cal Systems vol 32 no 2 pp 341ndash360 2012

[15] I Assani and K Presser ldquoA survey of the return timestheoremrdquohttparxivorgabs12090856

[16] P Zorin-Kranich ldquoCube spaces and the multiple term returntimes theoremrdquo Ergodic Theory and Dynamical Systems

[17] D Berend M Lin J Rosenblatt and A Tempelman ldquoMod-ulated and subsequential ergodic theorems in Hilbert andBanach spacesrdquo ErgodicTheory and Dynamical Systems vol 22no 6 pp 1653ndash1665 2002

[18] A Bellow and V Losert ldquoThe weighted pointwise ergodic the-orem and the individual ergodic theorem along subsequencesrdquoTransactions of the AmericanMathematical Society vol 288 no1 pp 307ndash345 1985

[19] J Bourgain ldquoPointwise ergodic theorems for arithmetic setsrdquoPublications Mathematiques de lrsquoInstitut des Hautes Etudes Sci-entifiques vol 69 no 1 pp 5ndash41 1989

[20] J Bourgain ldquoAn approach to pointwise ergodic theoremsrdquo inGeometric Aspects of Functional Analysis vol 1317 of LectureNotes in Mathematics pp 204ndash223 1988

[21] M Wierdl ldquoPointwise ergodic theorem along the prime num-bersrdquo Israel Journal of Mathematics vol 64 no 3 pp 315ndash3361988

[22] E Lesigne C Mauduit and B Mosse ldquoLe theoreme ergodiquele long drsquoune suite 119902-multiplicativerdquo Compositio Mathematicavol 93 no 1 pp 49ndash79 1994

[23] P Zorin-Kranich ldquoReturn times theorem for amenable groupsrdquohttparxivorgabs13011884

[24] B Host and A Maass ldquoNilsystemes drsquoordre 2 et parallelepipe-desrdquo Bulletin de la Societe Mathematique de France vol 135 no3 pp 367ndash405 2007

[25] I Assani E Lesigne and D Rudolph ldquoWiener-Wintner return-times ergodic theoremrdquo Israel Journal of Mathematics vol 92no 1ndash3 pp 375ndash395 1995

[26] A Leibman ldquoMultiple polynomial correlation sequences andnilsequencesrdquo Ergodic Theory and Dynamical Systems vol 30no 3 pp 841ndash854 2010

[27] V Bergelson B Host and B Kra ldquoMultiple recurrence andnilsequencesrdquo InventionesMathematicae vol 160 no 2 pp 261ndash303 2005

[28] HH SchaeferBanach Lattices and Positive Operators Springer1974

[29] T Eisner Stability of Operators and Operator Semigroups Oper-ator Theory Advances and Applications vol 209 BirkhauserBasel Switzerland 2010

[30] T Eisner B Farkas M Haase and R Nagel Operator TheoreticAspects of Ergodic Theory Graduate Texts in MathematicsSpringer to appear

[31] K Petersen Ergodic Theory vol 2 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1983

[32] Q Chu ldquoConvergence of weighted polynomial multiple ergodicaveragesrdquo Proceedings of the American Mathematical Societyvol 137 no 4 pp 1363ndash1369 2009


[33] W Parry ldquoErgodic properties of affine transformations andflows on nilmanifoldsrdquo American Journal of Mathematics vol91 pp 757ndash771 1969

[34] A Leibman ldquoPointwise convergence of ergodic averages forpolynomial sequences of translations on a nilmanifoldrdquo ErgodicTheory and Dynamical Systems vol 25 no 1 pp 201ndash213 2005

[35] M Rosenblum and J Rovnyak Topics in Hardy Classes and Uni-valent Functions Birkhauser Basel Switzerland 1994

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Algebra

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Stochastic AnalysisInternational Journal of


for every 119896 isin N and every nilsequence (119886119899) where the uni-

versal sets of convergence do not depend on (119886119899) This gener-

alised an earlier result byAssani et al [25] for sequences of theform (120582

119899

) 120582 isin T and 119896 = 2In this paper we search for good weights for ergodic

theorems using a functional analytic perspective and producedeterministic good weights We first introduce sequences ofthe form (⟨119879

119899

119909 1199091015840

⟩) which we call linear sequences if 119909 is in aBanach space119883 1199091015840 isin 119883

1015840 and 119879 is a linear operator on119883withrelatively weakly compact orbits see Section 2 later Using astructure result for linear sequences we show that they aregood weights for the multiple polynomial ergodic theorem(Section 4) and for the Wiener-Wintner type multiple returntime theorem discussed (Section 3) In the last section wepresent a counterexample showing that the assumption on theoperators cannot be dropped even for positive isometries onBanach lattices and the mean ergodic theorem

We finally remark that all results in this paper hold if wereplace linear sequences by a larger class of ldquoasymptotic nilse-quencesrdquo that is for sequences (119886

119899) of the form 119886

119899= 119887119899+ 119888119899

where (119887119899) is a nilsequence and (119888

119899) is a bounded sequence

satisfying lim119873rarrinfin

(1119873)sum119873

119899=1|119888119899| = 0 (cf Theorem 3)

Examples of asymptotic nilsequences (of step ge2 in general)aremultiple polynomial correlation sequences (119886

119899) of the form

119886119899= int119884

1198781199011(119899)

1198921sdot sdot sdot 119878119901119896(119899)

119892119896119889120583 (3)

for an ergodic invertible measure preserving system (119884 120583 119878)119896 isin N 119892


(119884 120583) and polynomials 119901119895with integer

coefficients 119895 = 1 119896 This follows from Leibman [26Theorem 31] and in the case of linear polynomials is due toBergelson et al [27 Theorem 19] Thus multiple polynomialcorrelation sequences provide another class of deterministicexamples of good weights for the Wiener-Wintner typemultiple return time theorem and the multiple polynomialergodic theorem discussed in Sections 3 and 4

2 Linear Sequences and Their Structure

A linear operator 119879 on a Banach space119883 has relatively weaklycompact orbits if for every 119909 isin 119883 the orbit 119879119899119909 119899 isin N

0 is

relatively weakly compact in119883

Definition 1 We call a sequence (119886119899) sub C a linear sequence if

there exists an operator119879 on a Banach space119883with relativelyweakly compact orbits and 119909 isin 119883 1199091015840 isin 119883

1015840 such that 119886119899=

⟨119879119899

119909 1199091015840

⟩ holds for every 119899 isin N

A large class of operators with relatively weakly compactorbits leading to a large class of linear sequences are powerbounded operators on reflexive Banach spaces Recall that anoperator119879 is called power bounded if it satisfies sup

119899isinN119879119899

lt

infin Another class of operators with relatively weakly compactorbits are power bounded positive operators on a Banachlattice 119871

1

(120583) preserving the order interval generated by astrictly positive function see for example Schaefer [28The-orem II510(f) and Proposition II83] See [29 Section I1]and [30 Section 161] for further discussion

Remark 2 By restricting to the closed linear invariant sub-space 119884 = lin119879119899119909 119899 isin N

0 induced by the orbit and using

the decomposition 1198831015840

= 1198841015840

oplus 1198841015840

0for 11988410158400= 1199091015840

1199091015840

|119884= 0 it

suffices to assume that only the relevant orbit 119879119899119909 119899 isin N0 is

relatively weakly compact in the definition of a linearsequence (⟨119879

119899

119909 1199091015840

⟩) Note that in this case 119879 has relativelyweakly compact orbits on 119884 by a limiting argument see forexample [29 Lemma I16]

Weobtain the following structure result for linear sequen-ces as a direct consequence of an extended Jacobs-Glicksberg-deLeeuw decomposition for operators with relatively weaklycompact orbits

Theorem 3 Every linear sequence is a sum of an almostperiodic sequence and a (bounded) sequence (119888

119899) satisfying

lim119873rarrinfin

(1119873)sum119873

119899=1|119888119899| = 0

Proof Let 119879 be an operator on a Banach space 119883 withrelatively weakly compact orbits By the Jacobs-Glicksberg-deLeeuw decomposition see for example [29 TheoremII48]119883 = 119883

119903oplus 119883119904 where

119883119903= lin 119909 119879119909 = 120582119909 for some 120582 isin T (4)

while every 119909 isin 119883119904

satisfieslim119873rarrinfin

(1119873)sum119873

119899=1|⟨119879119899

119909 1199091015840

⟩| = 0 for every 1199091015840

isin 1198831015840

(Recall that by the Koopman-von Neumann lemma seefor example Petersen [31 p 65] for bounded sequences thecondition lim


119873

119899=1|119888119899| = 0 is equivalent to

lim119895rarrinfin

119888119899119895

= 0 for some subsequence 119899119895 sub N with density

1)Let 119909 isin 119883 1199091015840 isin 119883

1015840 and define the sequence (119886119899) by 119886119899=

⟨119879119899

119909 1199091015840

⟩ For 119909 isin 119883119904we have lim


119873

119899=1|119886119899| = 0

by the aforementioned If now 119909 is an eigenvector corre-sponding to an eigenvalue 120582 isin T then 119886

119899= 120582119899

⟨119909 1199091015840

⟩Therefore for every 119909 isin 119883

119903 the sequence (119886

119899) is a uniform

limit of finite linear combinations of sequences (120582119899) 120582 isin T and is therefore almost periodic The assertion follows

3 A Wiener-Wintner Type Result forthe Multiple Return Time Theorem

In this section we show that one can take linear sequences asweights in the multiple Wiener-Wintner type generalisationof the return time theorem due to Zorin-Kranich [16] andAssani et al [25] discussed in the introduction

First we recall the definition of a property satisfieduniversally

Definition 4 Let 119896 isin N and 119875 be a pointwise property for 119896measure preserving dynamical systems We say that a prop-erty 119875 is satisfied universally almost everywhere if for everysystem (119884

1 1205831 1198781) and every 119892

1isin 119871infin

(1198841 1205831) there is a set

1198841015840

1sub 1198841of full measure such that for every 119910

1isin 1198841015840

1and

every system (1198842 1205832 1198782) for every system (119884

119896 120583119896 119878119896) and

119892119896isin 119871infin

(119884119896 120583119896) there is a set 1198841015840

119896sub 119884119896of full measure such

that for every 119910119896isin 1198841015840

119896the property 119875 holds




119895 119895 = 1 119896 of full measure



119899) satisfying

lim119873rarrinfin

(1119873)sum119873



1

119873

119873

sum

119899=1

1198861198991198921(119878119899

11199101) sdot sdot sdot 119892

119896(119878119899

119896119910119896)

1003816100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198921


1003817100381710038171003817infin

1

119873

119873

sum

119899=1

10038161003816100381610038161198861198991003816100381610038161003816

(5)


1015840

119896




119871infin



119899) given by

119886119899

= 119892(119878119899


1

119873

119873

sum

119899=1

1198861198991198781199011(119899)

11198921sdot sdot sdot 119878119901119896(119899)

1119892119896

(6)



1198781 every 119892

1 119892



1199011 119901









119899) satisfying

lim119873rarrinfin

(1119873)sum119873


Theorem 3

5 A Counter Example



is

119879 (1199051 1199052 ) = (0 119905

1 1199052 ) (7)


1015840

=

(119904119895) isin 1198831015840 we have

lim119873rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

⟨119879119899

119909 1199091015840

⟩ minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (8)


|119905119895| lt 120576 Then

for119873 isin N we have10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

⟨119879119899

119909 1199091015840

⟩ minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

infin

sum

119895=1

119905119895119904119899+119895

minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

119869

sum

119895=1

119905119895119904119899+119895

minus1

119873

119873

sum

119899=1

120582119899

119904119899

119869

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ 210038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119869

sum

119895=1

120582119895

119905119895

1

119873

119873+119895

sum

119899=1+119895

120582119899

119904119899minus

119869

sum

119895=1

120582119895

119905119895

1

119873

119873

sum

119899=1

120582119899

119904119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ 210038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

le

21198691199091

10038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

119873+ 2

10038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

(9)



119909 1199091015840


suminfin

119895=1119905119895

= 0 and for every 1199091015840



1





119899

119909 1199091015840






119899

119909 1199091015840








lim119903rarr1minus

119891 (119903120582) =

infin

sum

119895=1

120582119895

119905119895

= 0 (10)


⟨119879119899

119909 1199091015840


= (119904119895) isin 119897infin




119895) 997891rarr (120582

119895



infin


Acknowledgment


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119895 119895 = 1 119896 of full measure



119899) satisfying

lim119873rarrinfin

(1119873)sum119873



1

119873

119873

sum

119899=1

1198861198991198921(119878119899

11199101) sdot sdot sdot 119892

119896(119878119899

119896119910119896)

1003816100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171198921


1003817100381710038171003817infin

1

119873

119873

sum

119899=1

10038161003816100381610038161198861198991003816100381610038161003816

(5)


1015840

119896




119871infin



119899) given by

119886119899

= 119892(119878119899


1

119873

119873

sum

119899=1

1198861198991198781199011(119899)

11198921sdot sdot sdot 119878119901119896(119899)

1119892119896

(6)



1198781 every 119892

1 119892



1199011 119901









119899) satisfying

lim119873rarrinfin

(1119873)sum119873


Theorem 3

5 A Counter Example



is

119879 (1199051 1199052 ) = (0 119905

1 1199052 ) (7)


1015840

=

(119904119895) isin 1198831015840 we have

lim119873rarrinfin

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

⟨119879119899

119909 1199091015840

⟩ minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (8)


|119905119895| lt 120576 Then

for119873 isin N we have10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

⟨119879119899

119909 1199091015840

⟩ minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

infin

sum

119895=1

119905119895119904119899+119895

minus1

119873

119873

sum

119899=1

120582119899

119904119899

infin

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873

119873

sum

119899=1

120582119899

119869

sum

119895=1

119905119895119904119899+119895

minus1

119873

119873

sum

119899=1

120582119899

119904119899

119869

sum

119895=1

120582119895

119905119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ 210038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

=

10038161003816100381610038161003816100381610038161003816100381610038161003816

119869

sum

119895=1

120582119895

119905119895

1

119873

119873+119895

sum

119899=1+119895

120582119899

119904119899minus

119869

sum

119895=1

120582119895

119905119895

1

119873

119873

sum

119899=1

120582119899

119904119899

10038161003816100381610038161003816100381610038161003816100381610038161003816

+ 210038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

le

21198691199091

10038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

119873+ 2

10038171003817100381710038171003817119909101584010038171003817100381710038171003817infin

120576

(9)



119909 1199091015840


suminfin

119895=1119905119895

= 0 and for every 1199091015840



1





119899

119909 1199091015840






119899

119909 1199091015840








lim119903rarr1minus

119891 (119903120582) =

infin

sum

119895=1

120582119895

119905119895

= 0 (10)


⟨119879119899

119909 1199091015840


= (119904119895) isin 119897infin




119895) 997891rarr (120582

119895



infin


Acknowledgment


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















lim119903rarr1minus

119891 (119903120582) =

infin

sum

119895=1

120582119895

119905119895

= 0 (10)


⟨119879119899

119909 1199091015840


= (119904119895) isin 119897infin




119895) 997891rarr (120582

119895



infin


Acknowledgment


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article linear sequences and weighted ergodic theorems · 2019. 7. 31. · 4. weighted...

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