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Anoteonquasi-similarityofKoopmanoperators

K.Fr¡czekandM.Lema«czyk

Abstract

AnsweringaquestionofA.Vershikweconstructtwonon-weaklyisomorphicergodicautomor-

phismsforwhichtheassociatedunitary(Koopman)representationsareMarkovquasi-similar.We

alsodiscussmetricinvariantsofMarkovquasi-similarityintheclassofergodicautomorphisms.

1. Introduction

Markovoperatorsappearintheclassicalergodictheoryinthecontextofjoinings,seethe

monograph[7].Indeed,assumethatT i isanergodicautomorphismofastandardprobability

Borelspace(X i ,B i ,µ i ),i=1,2.ConsiderajoiningofT 1 andT 2 ,i.e.aT 1 ×T 2 -invariant

probabilitymeasureon(X 1 ×X 2 ,B 1 B 2 )withthemarginalsµ 1 andµ 2 respectively.Then

theoperator :L 2 (X 1 ,B 1 ,µ 1 )!L 2 (X 2 ,B 2 ,µ 2 )determinedby

h f 1 ,f 2 i L 2 (X 2 ,B 2 ,µ 2 ) =hf 1 1 X 2 , 1 X 1 f 2 i L 2 (X 1 ×X 2 ,B 1 B 2 ,) (1.1)

isMarkov(i.e.itisalinearcontractionwhichpreservestheconeofnon-negativefunctionsand

1 = 1 = 1 )andmoreover

U T 1 =U T 2 , (1.2)

whereU T i :L 2 (X i ,B i ,µ i )!L 2 (X i ,B i ,µ i )standsfortheassociatedunitaryoperator: U T i f=

fT i forf2L 2 (X i ,B i ,µ i ),i=1,2,whichisoftencalledaKoopmanoperator.Infact,each

Markovoperator:L 2 (X 1 ,B 1 ,µ 1 )!L 2 (X 2 ,B 2 ,µ 2 )satisfyingtheequivarianceproperty(1.2)

isoftheform forauniquejoiningofT 1 andT 2 (seee.g.[17],[24]).Markovoperators

correspondingtoergodicjoiningsarecalledindecomposable.

Inordertoclassifydynamicalsystemsoneusuallyconsidersthemeasure-theoreticisomor-

phism,i.e.theequivalencegivenbytheexistenceofaninvertiblemap S:(X 1 ,B 1 ,µ 1 )!

(X 2 ,B 2 ,µ 2 )forwhichST 1 =T 2 S.Themeasure-theoretic(metric)isomorphismimplies

spectralequivalenceoftheKoopmanoperators U T 1 andU T 2 ;indeed,U S −1 (whereU S −1 f 1 =

f 1 S −1 forf 1 2L 2 (X 1 ,B 1 ,µ 1 ))providessuchanequivalence.Theconversedoesnothold,

seee.g.[1];wealsorecallthatallBernoullishiftsarespectrallyequivalentwhiletheentropy

classifythemmeasure-theoretically[19].Onemayaskwhethertherecanbesomeothernatural

classi cationofdynamicalsystemswhichliesinbetweenmetricandspectralequivalence.

Given(X,B,µ)astandardprobabilityBorelspace,following[26],eachprobabilitymeasure

on(X×X,BB)withbothmarginalsµiscalledapolymorphism.Regardingautomorphisms

of(X,B,µ)asthecorrespondinggraphmeasures,in[26],Vershikoriginatesanewtheory

thetheoryofpolymorphisms inwhichpolymorphismsareanaloguesofautomorphismsof

(X,B,µ).Since,inviewof(1.1),thereisaone-to-onecorrespondencebetweenpolymorphisms

andMarkovoperatorsofL 2 (X,B,µ),asthecorrespondingequivalence(borrowedforoperator

theory,seebelow)VershikhaschosenMarkovquasi-similarity.Inparticular,Vershikproposed

2000MathematicsSubjectClassi cation37A05,37A30,37A35.

ResearchpartiallysupportedbyPolishMNiSzWgrantNN201384834;partiallysupportedbyMarieCurie

TransferofKnowledge EUprogram projectMTKD-CT-2005-030042(TODEQ)

Page2of15 K.FR � CZEKANDM.LEMA ‹ CZYK

tostudythisnewequivalencebetweenpolymorphismsandautomorphisms,andevenbetween

automorphismsthemselves.

RecallthatifA i isaboundedlinearoperatorofaHilbertspaceH i ,i=1,2,andifthereisa

boundedlinearoperatorV :H 1 !H 2 whoserangeisdenseandwhichintertwinesA 1 andA 2 ,

thisisVA 1 =A 2 V,thenA 2 issaidtobeaquasi-imageofA 1 (see[4]).Byduality,A 2 is

aquasi-imageofA 1 ifandonlyifthereexistsa1−1boundedlinearoperatorW:H 2 !H 1

intertwiningA 2 andA 1 .IfalsoA 1 isaquasi-imageofA 2 thenthetwooperatorsarecalled

quasi-similar.RecallalsothattwooperatorsA 1 andA 2 arequasi-a neifthereexistsa1−1

boundedlinearoperatorV :H 1 !H 2 withdenserangeintertwiningA 1 andA 2 .Ingeneral,

thenotionofquasi-a nityisstrongerthanquasi-similarity(seehoweverRemark2.2below).

AssumeadditionallythatA i isaMarkovoperatorofH i =L 2 (X i ,B i ,µ i ),i=1,2.IfA 2 is

aquasi-imageofA 1 andweadditionallyrequireV :L 2 (X 1 ,B 1 ,µ 1 )!L 2 (X 2 ,B 2 ,µ 2 )tobea

MarkovoperatorthenA 2 issaidtobeaMarkovquasi-imageofA 1 .IfadditionallyA 1 isa

Markovquasi-imageofA 2 thenthetwooperatorsarecalledMarkovquasi-similar.Operators

A 1 andA 2 areMarkovquasi-a neifthereexistsa1−1MarkovoperatorVbetweenthe

correspondingL 2 -spaceswithdenserangeintertwiningA 1 andA 2 .

NoticethateachKoopmanoperatorisalsoaMarkovoperator.Itisknown(seee.g.[15],[26])

thatifanintertwiningMarkovoperator:L 2 (X 1 ,B 1 ,µ 1 )!L 2 (X 2 ,B 2 ,µ 2 )isunitarythenit

hastobeoftheformU S whereSprovidesameasure-theoreticisomorphism.Ontheother

handthequasi-similarityofunitaryoperatorsimpliestheirspectralequivalence(seeSection2

below).Therefore,Markovquasi-similarityliesinbetweenthespectralandmeasure-theoretic

equivalenceofdynamicalsystems.OneofquestionsraisedbyVershikin[26]isthefollowing:

Dothereexisttwoautomorphismsthatarenotisomorphic

butareMarkovquasi-similar?

(1.3)

Inordertoanswerthisquestionnoticethatanyweaklyisomorphicautomorphisms(see[25])

T 1 andT 2 areautomaticallyMarkovquasi-similar;indeed,theweakisomorphismmeansthat

thereare 1 and 2 whicharehomomorphismsbetweenT 1 andT 2 andT 2 andT 1 respectively,

thenU 1 andU 2 yieldMarkovquasi-similarityof T 1 andT 2 .Hence,ifT 1 andT 2 areweakly

isomorphicbutnotisomorphic,weobtainthepositiveanswertothequestion(1.3).The rst

examplesofweaklyisomorphicbutnotisomorphicsystemsweregivenbyPolitin[21].For

furtherexampleswereferthereaderto[12],[13],[23],includingthecaseofK-automorphisms

[8].ItfollowsthatthenotionofMarkovquasi-similarityhastobeconsideredasaninteresting

re nementofthenotionofweakisomorphism,andinVershik’squestion(1.3)wehaveto

replace notisomorphic by notweaklyisomorphic .

Themainaimofthisnoteistoanswerpositivelythismodi edquestion(1.3)(see

Proposition4.4below).Wewouldliketoemphasizethatdespiteaspectral avorofthe

de nition,Markovquasi-similarityisfarfrombeingthesameasspectralequivalence.For

example,partlyansweringVershik’squestionraisedataseminaratPennStateUniversityin

2004whetherentropyisaninvariantofMarkovquasi-similarity,weshowthatzeroentropyas

wellasK-propertyareinvariantsofMarkovquasi-similarityofautomorphisms,whiletheyare

notinvariantsofspectralequivalenceofthecorrespondingunitaryoperators.Thesefactsand

relatedproblemswillbediscussedinSections5-7.

2.Quasi-similarityofunitaryoperatorsimpliestheirunitaryequivalence

AssumethatUisaunitaryoperatorofaseparableHilbertspaceH.Givenx2HbyZ(x)we

denotethecyclicspacegeneratedbyx,i.e.Z(x)=span{U n x: n2Z}.Wewilluseasimilar

notationZ(y 1 ,...,y k )forthesmallestclosedU-invariantsubspacecontainingy i ,i=1,...,k.

DenotebyTthe(additive)circle.ThentheFouriertransformofthe(positive)measure x

ANOTEONQUASI-SIMILARITYOFKOOPMANOPERATORS Page3of15

calledthespectralmeasureofx isgivenby

b x (n):= Z

e 2int d x (t)=hU n x,xiforeachn2Z.

Similarlythesequence(hU n x,yi) n2 Z istheFouriertransformofthe(complex)spectralmeasure

x,y ofxandy.Givenaspectralmeasurewedenote

H ={x2H: x }.

ThenH isaclosedU-invariantsubspacecalledaspectralsubspaceofH.

ItfollowsfromSpectralTheoremforunitaryoperators(seee.g.[11]or[20])thatthereisa

decomposition

H=H 1 H 2 ...

(2.1)

intospectralsubspacessuchthatforeachi1

n M

Z(x (i)

H i =

k ),

k=1

where i x (i)

x (i)

...(n i canbein nity),and i ? j fori 6=j.Theclass U ofall

nitemeasuresequivalenttothesum P i1 i isthencalledthemaximalspectraltypeofU.

Anotherimportantinvariantof U isthespectralmultiplicityfunctionM U :T!{1,2,...}[

{1}(see[11],[20])whichisde ned-a.e.,whereisanymeasurebelongingtothemaximal

spectraltypeofU.Notethatdecomposition(2.1)isfarfrombeinguniquebutif

n 0 M

1 M

Z(y (i)

H 0 i , H 0 i =

k )

i=1

k=1

isanotherdecomposition(2.1)inwhich i 0 i ,i1,thenn i =n 0 i fori1.Recallthatthe

essentialsupremumm U ofM U (calledthemaximalspectralmultiplicityofU)isequalto

inf{m1:Z(y 1 ,...,y m )=H forsomey 1 ,...,y m 2H}; (2.2)

ifthereisno good m,themm U =1.

AssumethatU i isaunitaryoperatorofaseparableHilbertspaceH i ,i=1,2.LetV :H 1 !

H 2 beaboundedlinearoperatorwhichintertwinesU 1 andU 2 .Thenforeachn2Zandx 1 2H 1

hU n 2 V x 1 ,V x 1 i=hU n 1 x 1 ,V V x 1 i,

sobyelementarypropertiesofspectralmeasures

Vx 1 = x 1 ,V Vx 1 x 1 .

(2.3)

AssumingadditionallythatIm(V)isdense,animmediateconsequenceof(2.3)isthatthe

maximalspectraltypeofaquasi-imageof U 1 isabsolutelycontinuouswithrespectto U 1 .It

isalsoclearthatgiveny (1) 1 ,...,y (1)

m 2H 1 wehave

V(Z(y (1) 1 ,...,y (1)

m ))=Z(V y (1) 1 ,...,V y (1)

m ).

Thisinturnimpliesthatthemaximalspectralmultiplicityofaquasi-imageof U 1 isatmost

m U 1 .

Proposition2.1. IfU 1 andU 2 arequasi-similarthentheyarespectrallyequivalent.

Proof.AssumethatV :H 1 !H 2 andW:H 2 !H 1 intertwineU 1 andU 2 andhavedense

ranges.Inviewof(2.3)bothoperators U 1 andU 2 havethesamemaximalspectraltypes.

Page4of15 K.FR � CZEKANDM.LEMA ‹ CZYK

Consideradecomposition(2.1)forU 1 :H 1 = L i1 H (1 i andletF i :=V(H (1)

)fori1.The

subspacesF i areobviouslyU 2 -invariantandlet (2 i (n (2 i )denotethemaximalspectraltype

(themaximalspectralmultiplicity)of U 2 onF i .Itfollowsfrom(2.3)that (2)

i (1 i for

i1and (2 i , (2 j aremutuallysingular(inparticular, F i ?F j )wheneveri 6=j.Moreover,

n (2)

i n (1 i ,i1.SinceVhasdenserange,H 2 = L i1 F i .Itfollowsthat(uptoequivalence

ofmeasures) P i1 (i) 2 isthemaximalspectraltypeofU 2 henceitisequivalentto P i1 (1)

andtherefore (1)

i (2 i fori1.Th esame reasoningappliedtothedecomposition H 2 =

L i1 F i andWshowsthatH 1 = L i1 W(F i )andthemaximalspectralty peofU 1 onW(F i )

isabsolutelycontinuouswithrespectto (2)

i (1 i ,i1.ItfollowsthatW(F i )=H (1) i for

alli1.Inparticular,wehaveprovedthatn (2)

i =n (1 i butweneedtoshowthatonF i the

multiplicityisuniform.Supposethisisnotthecase,i.e.thatforsomemeasure (2 i we

have

F i =Z(z 1 )...Z(z r )F 0 i ,

whereforj=1,...r, z j =,1r < n (2) i andthemaximalspectraltypeof U 2 onF 0 i is

orthogonalto.Wehave

=W(F i )=G i W(F 0 i ),

H (1)

where G i =W(Z(z 1 )...Z(z r ))andthemaximalspectraltypesonG i ,say(),and

W(F 0 i )aremutuallysingular.Itfollowsthatthemultiplicityof isatmostr,whichisa

contradictionsinceallmeasuresabsolutelycontinuouswithrespectto (1 i havemultiplicit y

n (1 i .

Remark2.2.Literallyspeaking,thenotionofquasi-similarityisweakerthanthenotion

ofquasi-a nity.Proposition3.4in[4]tellsusthatquasi-a neunitaryoperatorsareunitarily

equivalent.HenceProposition2.1showsinfactthatforunitaryoperatorsquasi-similarityand

quasi-a nityareequivalentnotions.

Itisnotclear(seeSection7)whetherthenotionsofMarkovquasi-similarityandMarkov

quasi-a nityofKoopmanoperatorscoincide.

3.Aconvolutionoperatorinl 2 (Z)

Inthissectionweproduceasequenceinl 2 (Z)whichwillbeusedtoconstructaMarkov

quasi-a nitybetweentwonon-weaklyisomorphicautomorphismsinSection4.

Denotebyl 0 (Z)thesubspaceofl 2 (Z)ofcomplexsequences¯x=(x n ) n2 Z suchthat{n2Z:

x n 6=0}is nite.

Proposition3.1.Thereexistsanonnegativesequence¯a=(a n ) n2 Z 2l 2 (Z)suchthat

P n2 Z a n =1and

forevery¯x=(x n ) n2 Z 2l 2 (Z)if¯a¯x2l 0 (Z)then¯x= ¯ 0. (3.1)

Eachelementy2l 2 (Z)isanL 2 -functiononZanditsFouriertransformisafunctionh2

L 2 (T)forwhichh(n)=y n foralln2Z.Moreover,theconvolutionofl 2 -sequencescorresponds

tothepointwisemultiplicationofL 2 -functionsonthecircle.Itfollowsthatinorderto ndthe

requiredsequence¯a,itsu cesto ndafunctionf2L 2 (T)suchthat

a n = ˆ f(n)0, P n2 Z a n =1;

ANOTEONQUASI-SIMILARITYOFKOOPMANOPERATORS Page5of15

foreveryg2L 2 (T),iff·g=0theng=0;

foreverynon-zerotrigonometricpolynomial P,ifP=f·gtheng /2L 2 (T).

Thisisdonebelow.

Lemma3.2. Iff:[0,1]!R + isaconvexC 2 -functionsuchthatf(1−x)=f(x)forall

x2[0,1]then ˆ f(n)0foralln2Z.

Proof.Byassumption, f 00 (x)0forall x2[0,1].Usingintegrationbypartstwice,for

n 6=0weobtain

ˆ f(n)= Z 1

f(x)e −2inx dx= Z 1

Z 1

f(x)cos(2nx)dx= 1

f(x)dsin(2nx)

Z 1

4 2 n 2 Z 1

=− 1

f 0 (x)sin(2nx)dx= 1

f 0 (x)dcos(2nx)

f 0 (1)−f 0 (0)− Z 1

= 1

4 2 n 2

f 00 (x)cos(2nx)dx

f 0 (1)−f 0 (0)− Z 1

4 2 n 2

|f 00 (x)cos(2nx)|dx

f 0 (1)−f 0 (0)− Z 1

4 2 n 2

f 00 (x)dx

=0.

ProofofProposition3.1.Letusconsiderf:[0,1]!Rde nedby

e − 1

|x−1/2| +2 if x 6=1/2

0 if x=1/2.

Sincef 00 (x)0forx2[0,1],byLemma3.2, a n = ˆ f(n)0.Asf:T!Risacontinuous

functionofboundedvariation,

f(x)=

1=f(0)= X

n2 Z

a n .

Sincef(x)6=0forx 6=1/2,iff·g=0forsomeg2L 2 (T)theng=0.

Suppose,contrarytoourclaim,thatthereexist g2L 2 (T)andanon-zerotrigonometric

polynomialPsuchthatf·g=P.Recallthatforeverym0wehave R 1

0 e 1/x x m dx=+1,

hence R 1

0 (e 1/x x m ) 2 dx=+1.SincePisanon-zeroanalyticfunction,thereexistsm0such

thatP (m) (1/2)6=0andP (k) (1/2)=0for0k < m.ByTaylor’sformula,thereexistC >0

and0< <1/2suchthat|P(x+1/2)|C|x| m forx2[−,].Itfollowsthat

|g(x)| 2 dx Z 1/2+

1/2

|P(x)| 2 /f(x) 2 dx= Z

|P(x+1/2)| 2 /f(x+1/2) 2 dx

(Cx m e 1/x ) 2 dx=+1,

andhenceg /2L 2 (T)whichcompletestheproof.

4.Twonon-weaklyisomorphicautomorphismswhichareMarkovquasi-similar

LetTbeanergodicautomorphismof(X,B,µ).AssumethatGisacompactmetricAbelian

groupwithHaarmeasure G .Ameasurablefunction':X!Giscalledacocycle.Usingthe

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