On the entangled ergodic theorem

On the entangled ergodic theorem

TANJAEISNER ANDD ´AVIDKUNSZENTI-KOVACS´

Abstract. We study the convergence of the so-called entangled ergodic averages 1

N^k XN n₁,...,n_k=1

Tm^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . .A1T₁^n↵(1),

wherekmand↵: {1, . . . ,m}!{1, . . . ,k}is a surjective map. We show that, on general Banach spaces and without any restriction on the partition↵, the above averages converge strongly as N ! 1under some quite weak compactness assumptions on the operatorsTj and Aj. A formula for the limit based on the spectral analysis of the operatorsTj and the continuous version of the result are presented as well.

Mathematics Subject Classification (2010): 47A35 (primary); 37A30 (sec- ondary).

1. Introduction

The classical mean ergodic theorem has inspired many mathematicians and led to several generalisations and extensions. We mention Berend, Lin, Rosenblatt, Tem- pelman [3] for modulated and subsequential ergodic theorems ande.g.Kra [13] for an overview on multiple ergodic theorems as well as for the history of the subjects and further references.

In this note we study a further extension of the mean ergodic theorem, namely the so-called entangled ergodic theorem. Let↵ : {1, . . . ,m}!{1, . . . ,k}be a sur- jective map for some positive integerskm, and takeT1, . . . ,TmandA1, . . . ,Am 1

to be linear operators on a Banach space X. We investigate the convergence of the entangled Ces`aro means

1 N^k

XN n1,...,nk=1

Tm^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . .A1T₁^n↵(1). (1.1) This type of ergodic theorems was introduced by Accardi, Hashimoto, Obata [1]

motivated by quantum stochastics and was then studied by Liebscher [15] and Received December 16, 2010; accepted in revised form June 4, 2011.

F. Fidaleo [9, 11]. In their studies, T1 = . . . = Tm =: U is a unitary operator on a Hilbert space. Besides a technical assumption of Liebscher [15], there were two basic situations in which weak or strong convergence of the entangled ergodic aver- ages could be proved. The first is when A_j are arbitrary and the unitary operatorU is almost periodic (see Definition 2.1 below), see Liebscher [15] and Fidaleo [11].

The second is when the operators Aj are compact andU is arbitrary unitary, see Fidaleo [9].

In this paper we consider a more general situation. We assume the operators Tj belong to a large class including power bounded operators on reflexive Banach spaces. Further, we require a quite general compactness condition on the pairs (Aj,Tj) generalising both of the above cases. More precisely, we make the fol- lowing assumptions.

(A1) (Weakly compact orbits ofTj)

The operator Tm is power bounded and totally mean ergodic and each of T1, . . . ,Tm 1has relatively weakly compact orbits,i.e.,{T_jⁿx : n2N}is rela- tively compact inXin the weak topology for everyx 2Xand 1 j m 1.

(A2) (Joint compactness of(Aj,Tj))

Every A_j is compact on the orbits of T_j,i.e.,{A_jT_jⁿx : n 2 N}is relatively compact in Xfor everyx2 Xand 1 j m 1.

Recall that an operator T is called power bounded if sup_n2NkTⁿk < 1. More- over, T is called totally mean ergodic if the operators T are mean ergodic for every 2T^,Tthe unit circle. Every operator with relatively weakly compact or- bits is automatically totally mean ergodic. Note that assumption (A1) is not very restrictive. As mentioned above, every power bounded operator on a reflexive Ba- nach space has relatively weakly compact orbits by the Banach–Alaoglu theorem.

Another important class of examples is given by power bounded positive opera- tors on a Banach lattice L¹(µ) preserving an order interval generated by a strictly positive function, seee.g.Schaefer [18, Theorem II.5.10(f) and Proposition II.8.3]

or [7, Section I.1] for further information. While forming a large class, operators with relatively weakly compact orbits admit good asymptotic properties, see Theo- rem 3.1 below.

Under the assumptions (A1) and (A2) we show that the entangled Ces`aro means (1.1) converge strongly and describe their limit operator, see Theorem 3.4.

It turns out that only specially interacting projections corresponding to unimodular eigenvalues ofTj (in combination with the operators Aj) contribute to the limit.

The paper is organised as follows. We first treat the special case assuming that all but the lastTj are almost periodic (Section 2). Here we show how to reduce the problem to the case whenT1=. . .=Tm 1andA1=. . .= Am 1. In the context of Hilbert spaces and pair partitions, this case was considered recently in Fidaleo [11].

In Section 3 the general case is treated. We further discuss the connection to non- commutative multiple ergodic theorems, them being an important recent field of

research. In Section 5, we finally study the case of strongly continuous semigroups and the strong convergence of the ergodic averages

1 t^k

Z

[0,t]^k

Tm(s↵(m))Am 1Tm 1(s↵(m 1))Am 2. . .A1T1(s↵(1))ds1. . .dsk. Note that the study of convergence of entangled ergodic averages for the continuous time scale seems to be new.

Before proceeding we first show by an example that one cannot drop assump- tion (A2) even whenm =2 and the operatorT1= T2 =:U is unitary and weakly stable,i.e., satisfies limn!1Uⁿ =0 in the weak operator topology.

Proposition 1.1. There exists a Hilbert spaceH, a weakly stable unitary operator U onH and an operatorA2L^(H)such that the Ces`aro means

1 N

XN n=1

UⁿAUⁿ

do not converge weakly.

Proof. Let H := `<sup>2</sup>(Z<sup>)</sup>and consider the standard orthonormal base {e<sub>b</sub>}b2Z. Let thenUbe the unitary left shift operator. Let further(f(n))<sub>n2N</sub>+be a 0 1 sequence that is not Ces`aro-summable, and define Aby

Ae_b:=

⇢ef( b) b wheneverb<0, eb wheneverb 0.

Then A is a bounded operator, and we have UⁿAUⁿe0 = ef(n) for alln 2 N^. HencehUⁿAUⁿe₀,e₀i=1 f(n), and since(f(n))_n2N+ is not Ces`aro-summable, the Ces`aro-means _N¹ PN

n=1UⁿAUⁿcannot converge weakly either.

ACKNOWLEDGEMENTS. The authors are grateful to Rainer Nagel and Marco Schreiber for valuable discussions and comments.

2. The almost periodic case

We first consider the situation when all operatorsTj are almost periodic. In the case when↵is a pair partition,Xis a Hilbert space and the operatorsT_j are all equal to a unitary operator, this has been studied by Liebscher [15] and Fidaleo [11].

Definition 2.1. An operatorT 2L^(X)acting on a Banach space is calledalmost periodicif it is power bounded and satisfies

X =lin x 2X 9 2T:T x = x .

The following result is the first step towards the general situation.

Theorem 2.2. Let X be a Banach space,T₁, . . . ,T_{m 1}be almost periodic opera- tors onX,Tm2L^(X)be power bounded and totally mean ergodic,A1, . . . ,Am 12 L^{(X), and↵} : {1, . . . ,m} ! {1, . . . ,k}be surjective for somek  m. Then the entangled Ces`aro means

1 N^k

XN n1,...,nk=1

T_m^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . .A1T₁^n↵(1) (2.1) converge strongly asN ! 1.

Proof. We shall proceed by induction onm, giving an explicit form for the limit.

First recall that any operator with relatively weakly compact orbits (in particu- lar any almost periodic operator) is mean ergodic, seee.g.[7, Theorem I.2.9], and to have relatively weakly compact orbits is an invariant property under multiplication by a unimodular constant. We may therefore introduce the mean ergodic projections

P^(j):= lim

N!1

1 N

XN n=1

( ¹Tj)ⁿ, where 1 j mand| | =1.

We now show that the limit of the entangled Ces`aro means (2.1)is given by the formal sum

X

j2 j(1jm) Q

i2↵ 1(a) i=1(1ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .A1P⁽¹⁾₁ . (2.2)

Here, j denotes the point spectrum ofTj (1 j m) and (2.2) should be under- stood as the strong limit of the net

8>

>>

<

>>

>:

X

j2Fj(1jm) Q

i2↵ 1(a) i=1(1ak)

P^(m)_m A_{m 1}P^(m_{m 1}¹⁾A_{m 2}. . .A₁P⁽¹⁾₁ 9>

>>

=

>>

>;

Fj⇢ jfinite(1jm)

.

As seen above, this holds form=1. Suppose we know that it holds for any choice of(m 2)almost periodic operatorsTj and(m 2)bounded operatorsAi(m 2).

We may suppose without loss of generality that↵(1)=1.

If now↵ ¹(1)= {1}, then (2.1) can be written as 1

N^{(k 1)} XN n2,...,nk=1

T_m^n↵(m)A_{m 1}T_m^n↵(m₁ ¹⁾A_{m 2}. . .A₂T₂^n↵(2)

! A₁ 1

N XN n1=1

T₁ⁿ¹

! .

By the induction hypotheses and the joint continuity of multiplication in the strong operator topology, this expression converges to

0 BB B@

X

j2 j(2jm) Q

i2↵ 1(a) i=1(2ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .A2P⁽²⁾₂ 1 CC CAA1P₁⁽¹⁾

= X

j2 j(1jm) Q

i2↵ 1(a) i=1(1ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .A1P⁽¹⁾₁ .

If, on the other hand, there exists an 1 < l  m with↵(l) = 1, then consider an eigenvector x 2 X ofT1pertaining to some eigenvalue 2 T. We can rewrite the entangled means applied toxas

1 N^k

XN n1,...,nk=1

Tm^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . .A1T₁^n↵(1)x

= 1 N^k

XN n1,...,nk=1

T_m^n↵(m)A_{m 1}T_m^n↵(m₁ ¹⁾A_{m 2}. . .A₁ ^n↵(1)x

= 1 N^k

XN n1,...,nk=1

Tm^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . . ( Tl)^n↵(l). . .A2T₂^n↵(2)(A1x).

This reduces the problem to the(m 2)almost periodic operatorsT₂, ..., T_l, ...,T_m, and(m 2)bounded operatorsA₂, . . . ,A_{m 1}. The induction hypotheses together withx= P⁽¹⁾x yields that the limit is

X

j2 j(2jm,j6=l) l2 l Q

i2↵ 1(a)\{1} i=1(1ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .P^(l)1

l. . .A2P⁽²⁾₂ (A1x)

= X

j2 j(2jm) 1= Q

i2↵ 1(a) i=1(1ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .P^(l)_l . . .A2P⁽²⁾₂ A1P⁽¹⁾₁ x

= X

j2 ^j(1jm) Q

i2↵ 1(a) i=1(1ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .A1P⁽¹⁾₁ x.

Since the eigenvectors ofT1corresponding to unimodular eigenvalues spanX and the entangled Ces`aro means are uniformly bounded, we obtain the convergence on

Xto the required limit.

The following proposition shows that for the study of convergence of the en- tangled ergodic averages, it may be assumed without loss of generality that all op- eratorsA_j are equal, and the same for the operatorsT_j (1 j m 1). This trick will be used later, to simplify the proofs of Theorem 3.4 and Theorem 5.5.

Proposition 2.3. Suppose we know that Theorem2.2and the form of the limit given by (2.2)hold under additional assumption T1 = . . . = Tm 1 and A1 = . . . =

Am 1. Then it also holds in full generality.

Proof. Consider the spaceX :=X^mwith the diagonal operators T := diag(T1,T2, . . . ,Tm 1,I)2L⁽X^),

S := diag(I,I, . . . ,I,Tm)2L⁽X⁾ and the off-diagonal operator

A:=(( _{a 1,b}A_a))_a,b2L⁽X^),

where is the Kronecker symbol. Since all theTj’s (1  j  m 1) are almost periodic, so isT^{, and}Sis clearly totally mean ergodic onX. We can hence apply the weaker statement of Theorem 2.2 to the operatorsT^,S ^andA. For the vector (x,0, . . . ,0)^T 2X ^{for somex}2 X, this yields the existence of

Nlim!1

1 N^k

XN n1,...,nk=1

S^n↵(m)AT^{n↵(m 1)}A^{. . .}AT^{n↵(1)(x,0, . . . ,0)T. (2.3)} However, each of the summands has the form

(0, . . . ,0,Tm^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . .A1T₁^n↵(1)x)^T,

and hence the convergence of(2.3)in the last coordinate implies the required con- vergence of(2.1). Concerning the explicit form of the limit in question, the expres- sion(2.2)yields that it is the last coordinate of

X

j2 (1jm) Q

i2↵ 1(a) i=1(1ak)

P^S_mAP^T_{m 1}^{. . .}AP^T₁^{(x,0, . . . ,0)T, (2.4)}

where P^{S and}P^T are the mean ergodic projections onto the eigenspace corre- sponding to ofS^andT, respectively. Due to the diagonality ofT ^andS^{, each of} the components inX = X^m is T^{- and}S-invariant, hence we in fact haveP^T = diag(P⁽¹⁾, . . . ,P^{(m 1)},1_{1}( )I) and P^S = diag(1_{{1}( )I}, . . . ,1_{{1}( )I},P^(m)) where 1_M denotes the charasteristic function of a set M. The summands in (2.4) thus have the form

(0, . . . ,0,P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .A1P⁽¹⁾₁ x)^T.

Taking into consideration that the mean ergodic projections satisfy RanP^(j) = y2X|Tjy = y , and hence P^(j) = 0 whenever 62 ^j, the limit reduces to the required form.

3. The general case

We now extend the results from the previous section to the case when only the orbits along the family of operators A_iT_iⁿ _n2N+ are relatively compact for all 1 i  m 1,i.e., assuming (A2).

The key for our considerations will be the following extended version of a clas- sical decomposition theorem, seee.g.Krengel [14, Section 2.2.4] or [7, Theorem II.4.8].

Theorem 3.1 (Jacobs-Glicksberg-de Leeuw decomposition). LetXbe a Banach space and letT 2L^(X)have relatively weakly compact orbits. ThenX= Xr Xs, where

Xr:=lin{x2 X|T x= x for some 2T}, Xs:=

⇢

x2X| lim

j!1T^njx=0weakly for some sequence{nj}¹j=1with density1 , with both subspaces being invariant under T. In addition, ifX⁰is separable, then there exists a sequence {n_j}¹_j=1 with density 1 such that lim_j!1T^nj|Xs = 0 weakly.

Recall that the density of a set M⇢Nis defined by d(M)= lim

n!1

|M\{1, . . . ,n}|

n 1,

whenever the above limit exists.

We further need the following well-known facts.

Lemma 3.2 (Koopman-von Neumann). For a bounded sequence{an}¹_n=1⇢[0,1) the following assertions are equivalent.

(a) lim

n!1

1 n

Xn k=1

ak =0.

(b) There exists a subsequence{nj}¹_j=1ofNwith density1such that lim

j!1anj=0.

We refer toe.g.Petersen [17, page 65] for the proof.

Lemma 3.3. Let X be a Banach space and let{Tn}¹_n=1,{Sn}¹_n=1 ⇢ L⁽X). If both {Tnx : n 2 N}and{Snx : n 2N}are relatively compact inX for everyx 2 X, then so is{TnSnx : n2N}for everyx 2X.

Proof. Since compact sets are bounded and by the uniform boundedness principle, there exists M > 0 such thatkTnk  M andkSnk  M holds for everyn 2 N^+. Take nowx 2 X and a sequence{nj} ⇢N. Then there exists a subsequence{mj} of{nj}such that limj!1Smjx = yfor some y 2 X. Furthermore, there exists a

subsequence of{mj}which we again denote by{mj}such that limj!1Tmjy = z for somez2 X. This yields

kTmjSmjx zk MkSmjx yk+kTmjy zk !0 as j ! 1, proving relative compactness of{TnSnx : n2N}.

The following is the main result of the paper.

Theorem 3.4. Let Xbe a Banach space and↵ : {1, . . . ,m} ! {1, . . .k}be sur- jective for some k  m. Let further T1, . . . ,Tm,A1, . . . ,Am 1 2 L^(X)satisfy assumptions(A1)and(A2). Then the entangled ergodic averages

1 N^k

XN n1,...,nk=1

Tm^n↵(m)Am 1T_m^n↵(m₁ ¹⁾Am 2. . .A1T₁^n↵(1)

converge strongly, and their limit is given by X

j2 j(1jm) Q

i2↵ 1(a) i=1(1ak)

P^(m)_m Am 1P^(m_{m 1}¹⁾Am 2. . .A1P⁽¹⁾₁ ,

where j = P (Tj)\T^andP(_j^j)is the projection onto the eigenspace ofTj corre- sponding to j,i.e., the mean ergodic projection of the operator jTj.

Proof. As in the proof of Corollary 2.3 we may assumeTj = T and Aj = Afor someT,A2L^{(X)and all 1} j m 1. Forx 2X, we have to show convergence of

1 N^k

XN n1,...,nk=1

Tm^n↵(m)AT^{n↵(m 1)}A. . .AT^n↵(1)x. (3.1) By Theorem 3.1, the summands in (3.1) satisfy

Tm^n↵(m)AT^{n↵(m 1)}A . . . AT^n↵(1)x

=

mX1 a=1

Tm^n↵(m)A. . .AT^n↵(a)PsAT^{n↵(a 1)}PrA. . .AT^n↵(1)Prx

+T_m^n↵(m)AT^{n↵(m 1)}P_rA. . .AT^n↵(1)P_rx,

where Pr andPsare the projections ontoXrandXspertaining toT from Theorem 3.1, respectively. By Theorem 2.2, the averages of the second summand above con- verge to the desired limit. It remains to show that the averages of the first summand converge to 0,i.e., that for everyx2 Xand 1am 1 one has

Nlim!1

1 N^k

XN n1,...,nk=1

Tm^n↵(m)AT^{n↵(m 1)}A...AT^n↵(a)PsAT^{n↵(a 1)}PrA...AT^n↵(1)Prx=0.

(3.2)

Consider

K:= {AT^{na 1}PrA. . .ATⁿ¹Prx|na 1, . . . ,n1 2N}

which is relatively compact by assumption and Lemma 3.3(a). We now show that the dual space of the smallest T-invariant subspace Y containing K is separa- ble. Observe first that Y = lin{Tⁿx|n 2 N^{, x} 2 K}. We first show that the set Orb(K) := {Tⁿx|n 2 N^{, x} 2 K}is relatively weakly compact. Take a sequence {T^njx_j}¹j=1withx_j 2K andn_j 2N^{. SinceK} is relatively compact, there exists a subsequence of{xj}(which we again denote by{xj}) converging to somez. More- over, sinceT has relatively weakly compact orbits, there is a subsequence of{nj} (which we again denote by {nj}) such that limj!1T^njz = wweakly for some w2X. So we have

|hT^njxj w,yi||hT^njxj T^njz,yi| + |hT^njz w,yi|!0 8y2X⁰, i.e., limj!1T^njxj = wweakly, and therefore Orb(K)is relatively weakly com- pact. Since Y is separable, and the weak topology is metrisable on weakly com- pact subsets of separable spaces (seee.g.Dunford, Schwartz [5, Theorem V.6.3]), the weak topology on Orb(K)is metrisable. So it is induced by countably many {yn}⇢Y⁰, implying separability ofY⁰byY =linOrb(K).

Theorem 3.1 assures now the existence of a sequence{nj}⇢Nwith density 1 such that limj!1T^njPsy=0 weakly for everyy2 K. This implies

jlim!1AT^njPsy=0 weakly for everyy2K. (3.3) We now show that

jlim!1kAT^njPsyk=0 uniformly iny2K. (3.4) Indeed, assume that for some y2K,AT^njPsydoes not converge strongly to zero.

Then there exist > 0 and a subsequence{mj}of{nj}such thatkAT^mjPsyk for every j. By relative compactness of {ATⁿPsy : n 2 N}, the sequence {AT^mjPsy}¹_j=1has a strong accumulation point which by (3.3) must be zero, a con- tradiction. Thus, limj!1kAT^njPsyk=0 for everyy 2K and therefore uniform in y 2 K, since strong convergence inL^(X)implies uniform strong convergence on compact sets. Since the sequence{nj}has density 1, the equation (3.2) follows from (3.4) and Lemma 3.2.

4. Connection to convergence of multiple ergodic averages

In this section we discuss connection between the considered above entangled er- godic theorems and the important topic as multiple ergodic averages. The latter is concerned with non-commutative dynamical systems and was studied by Niculescu,

Str¨oh and Zsid´o [16], Duvenhage [6], Beyers, Duvenhage and Str¨oh [4], Fida- leo [10], and Austin, Eisner, Tao [2].

We first introduce what we mean by a non-commutative dynamical system and corresponding multiple ergodic averages.

Definition 4.1. A von Neumann (or non-commutative) dynamical system is a triple (M^,⌧, ), whereMis a von Neumann algebra,⌧ :M! Cis a faithful normal trace, and :M!M^{is a⌧}-preserving⇤-automorphism. We say fork2N^that the multiple ergodic averages

1 N

XN n=1

n(a1) ²ⁿ(a2) . . . ^kn(ak) (4.1)

convergestronglyif they converge in the⌧-norm defined bykak⌧ :=p

⌧(aa^⇤). The averages in (4.1) are calledweaklyconvergent if

1 N

XN n=1

⌧(a0 n(a1) ²ⁿ(a2)· · · ^kn(ak)) converges as N ! 1for everya02M^.

We recall that by the Gel’fand–Neumark–Segal theory, Mcan be identified with a dense subspace of a Hilbert space, where the Hilbert spaceHcan be obtained as the completion ofMwith respect to the⌧-norm. Thus, identifying elements of Mwith elements inH and by the standard density argument, strong convergence of the multiple ergodic averages (4.1) corresponds to norm convergence in H and weak convergence of (4.1) corresponds to weak convergence in H.

Recall further that for the automorphism there exists a unitary operatoru 2 L^{(H)such that (a)}= uau ¹, seee.g.[12, Proposition 4.5.3]. (Note thatu does not necessarily belong toM.) Thus, averages (4.1) take the form

1 N

XN n=1

uⁿa1uⁿa2· · ·uⁿaku ^kn, (4.2) i.e., are a special case of entangled ergodic averages for the constant partition

↵(j)=1 for every j 21, . . . ,kand the operatorsu, . . . ,u,u ^k.

It is well-known that strong (weak) topology and strong (weak) operator topol- ogy on Mcoincide on bounded sets. Therefore, there is a direct correspondence between strong (weak) convergence of multiple ergodic averages (4.1) and strong (weak)operatorconvergence of the entangled ergodic averages (4.2),cf.also Fida- leo [10].

Proposition 4.2. Let(M^{,⌧, )}be a von Neumann dynamical system and H and uas above. Let furthera1, . . . ,ak 2M. Then the multiple ergodic averages(4.1) converge strongly (weakly) if and only if the entangled averages(4.2)converge in the strong(weak)operator topology.

As was shown in [2], multiple ergodic averages (4.1) do not converge in general fork 3. Theorem 3.4 shows now that for every von Neumann dynamical system there is a classKdepending on the system such that the multiple ergodic averages converge strongly whenevera₁, . . . ,a_k 2K^{, and}Kcan be chosen as the subspace of all elementsa 2M^{such that}{auⁿ :n 2N}is relatively compact inL^(H)for the strong operator topology.

Remark 4.3. Note that also more general sequences of powers than arithmetic se- quences for the multiple ergodic averages can be treated by a slight modification in the last inductive step of the proof of Theorem 2.2. For instance, consider the averages

1 N^k

XN n1,...,nk=1

n↵(1)(a₁) ^{n↵(1)+n↵(2)}(a₂)· · ·

Pm

j=1n↵(j)(a_m). (4.3) These can then be rewritten as

1 N^k

XN n1,...,nk=1

u^n↵(1)a₁u^n↵(2)a₂· · ·u^n↵(m)a_mu ^Pmj=1n↵(j). (4.4) The last exponent being a sum ofn↵(j)’s rather than a single one does not matter on the almost weakly stable part, the compactness arguments of the proof of Theo- rem 3.4 still hold. On the almost periodic part however, supposex is an eigenvector to the unimodular eigenvalue . Then

1 N^k

XN n1,...,nk=1

u^n↵(1)a1u^n↵(2)a2· · ·u^n↵(m)amu ^Pmj=1n↵(j)x

= 1 N^k

XN n1,...,nk=1

u^n↵(1)a1u^n↵(2)a2· · ·u^n↵(m)am Pm

j=1n↵(j)x,

and the powers of the eigenvalue have to be pulled forward to not only a single operator as in the inductive proof of Theorem 2.2, but distributed amongst all of them to get back to the standard form:

1 N^k

XN n1,...,nk=1

u^n↵(1)a₁u^n↵(2)a₂· · ·u^n↵(m)a_m ^Pmj=1n↵(j)x

= 1 N^k

XN n1,...,nk=1

( ¹u)^n↵(1)a1( ¹u)^n↵(2)a2· · ·( ¹u)^n↵(m)amx.

Hence the averages (4.3) converge strongly if{ajuⁿ :n2N}is relatively compact inL^(H)for the strong operator topology for every 1 j m.

5. Continuous case

In this section we treat the continuous time scale, where the operatorsT_j and their powers are replaced by strongly continuous (C₀-) semigroups(T_j(t))_{t 0}. The study of the continuous version for entangled ergodic averages seems to be new. Some steps in the proofs are similar to the discrete case and will be skipped. For the general theory of strongly continuous semigroups we refer toe.g.Engel, Nagel [8].

For a semigroup(T(t))_{t 0}we often writeT(·).

Definition 5.1. AC0-semigroup of operators(T(t))t 0 ⇢ L^(X)acting on a Ba- nach space X is calledalmost periodicif it is bounded (i.e.sup_{t 0}kT(t)k < 1) and satisfies

X =linn

x 2X 9'2R:T(t)x =e^i'tx8t 0o .

Recall that by the spectral mapping theorem (seee.g.[8, Corollary IV.3.8]),T(t)x= e^i'txfor everyt 0 if and only ifBx =i'x for the generatorBofT(·).

We again treat the almost periodic case first.

Theorem 5.2. LetXbe a Banach space,T₁(·), . . . ,T_{m 1}(·)be almost periodicC₀- semigroups on X,Tm(·)a bounded totally mean ergodicC0-semigroup on X and

A1, . . . ,Am 12L(X). Then the entangled Ces`aro means 1

t^k Z

[0,t]^k

Tm(s↵(m))Am 1Tm 1(s↵(m 1))Am 2. . .A1T1(s↵(1))ds1 . . .dsk (5.1) converge strongly ast ! 1.

The integrals in (5.1) are defined strongly. Recall that a semigroup (T(t))_{t 0} is called totally mean ergodic if the semigroup(e^i'tT(t))_{t 0}is mean ergodic for every '2R^.

Proof. Since almost periodicC0-semigroups are totally mean ergodic by the stan- dard density argument, we can for each Tj(·) (1  j  m) define the projec- tions P'^(j) as the mean ergodic projections of (e ^i'tT_j(t))_{t 0} with range {y 2 X|T_j(t)y = e^i'ty8t 0}. Let _j denote the point spectrum of the generator Bj of the semigroupTj(·)oniR⁽¹ j m).

The proof can then be concluded by an induction argument onm analogous to the discrete case showing that the limit of the entangled Ces`aro means(5.1)is given by

X

'j2 j(1jm) P

i2↵ 1(a)'i=0(1ak)

P_'^(m)_m A_{m 1}P_'^(m_{m 1}¹⁾A_{m 2}. . .A₁P_'⁽¹⁾₁ . (5.2)

We shall use the following analogue of Theorem 3.1, seee.g.[7, Theorem III.5.7].

Theorem 5.3 (Continuous Jacobs-Glicksberg-de Leeuw decomposition). LetX be a Banach space and letT(·)⇢L^{(X)be aC}0-semigroup with relatively weakly compact orbits,i.e., such that{T(t)x : t 2 [0,1)}is relatively compact in X in the weak topology for everyx2 X. ThenX = Xr Xs, where

Xr = linn

x 2 X|T(t)x=e^i'tx8t 0for some'2R^o, Xs =

⇢

x 2 X| lim

M3j!1T(t)x =0weakly for someM ⇢[0,1)with density1 , with both subspaces being invariant under T. In addition, ifX⁰is separable, then there exists a set M ⇢ [0,1) with density1such that lim_M3t!1T(t)|Xs = 0 weakly.

The density of a setM ⇢[0,1)is defined by d(M)= lim

t!1

(M\[0,t]) t 1,

with (·)denoting the Lebesgue measure, whenever the above limit exists.

We further need the following continuous version of Lemma 3.2.

Lemma 5.4 (Koopman-von Neumann, continuous version). For a continuous function f : [0,1)![0,1)the following assertions are equivalent.

(a) lim

t!1

1 t

Z

[0,t]

f(s)ds=0.

(b) There exists a subsetMof[0,1)with density1such that lim

s2M,s!1 f(s)=0.

For the proof, which is analogous to the discrete case, seee.g.[7, Lemma III.5.2].

The main result of this section is the following:

Theorem 5.5. LetXbe a Banach space,m2N^,Tm(·), ...,Tm(·)beC0-semigroups on X, A1, . . . ,Am 1 2 L^{(X), and↵} : {1, . . . ,m} ! {1, . . . ,k}be a surjective map for somek,m 2N. Assume the following.

(A1c) The semigroup Tm(·) is bounded and totally mean ergodic andTj(·) has relatively weakly compact orbits for every1 j m 1.

(A2c) Every Aj is compact on the orbits ofTj(·),i.e.,{AjTj(t)x : t 2[0,1)}is relatively compact inXfor everyx2 Xand1 j m 1.

Then the entangled ergodic averages 1

t^k Z

[0,t]^k

Tm(s↵(m))Am 1Tm 1(s↵(m 1))Am 2. . .A1T1(s↵(1))ds1 . . .dsk

converge strongly. Denoting the generator ofTj(·)by Bj (1 j m), the strong limit is given by the formula

X

i'j2 j(1jm) P

j2↵ 1(a)'j=0(1ak)

P_'^(m)_m A_{m 1}P_'^(m_{m 1}¹⁾A_{m 2}. . .A₁P_'⁽¹⁾₁ ,

where j = P (Bj)\iR^andP'(j^j)is the projection onto the eigenspace ofBj corre- sponding toi'_j, i.e., the mean ergodic projection of the semigroup(e ^i'jtTj(t))_{t 0}. Proof. Using the arguments from Proposition 2.3, we may again assume that we haveTj(·)=T(·)andAj = Afor 1 j m 1. It is to be shown that

1 t^k

Z

[0,t]^k

Tm(s_↵(m))AT(s_{↵(m 1)})A. . .AT(s_↵(1))x ds1 . . .dsk (5.3) converges for every x 2 X. By Theorem 5.3, the integrand can be split with the help of the projections PrandPsontoXr andXs, respectively, and we have

Tm(s↵(m))AT(s↵(m 1))A. . .AT(s↵(1))x

=

mX1 a=1

Tm(s_↵(m))A. . .AT(s_↵(a))PsAT(s_{↵(a 1)})PrA. . .AT(s_↵(1))Prx

+ Tm(s↵(m))AT(s↵(m 1))PrA. . .AT(s↵(1))Prx.

The integral means of the second term converge by Theorem 5.2 to the desired limit, and so it is enough to show that the rest converges in mean to 0,i.e.for everyx 2X and 1am 1 one has

tlim!1

1 t^k

Z

[0,t]^k

T_m(s_↵(m))A. . .AT(s_↵(a))P_sAT(s_{↵(a 1)})P_rA. . .AT(s_↵(1))P_rx =0.

(5.4) Consider

K := {AT(sa 1)PrA. . .AT(s1)Prx|sa 1, . . . ,s12[0,1)}.

This set is relatively compact by Lemma 3.3 and the assumption. As in the discrete case, one can show that the dual space of the smallest T(·)-invariant subspace Y containing K is separable. Note that the separability ofY itself follows from the strong continuity of the semigroup, as it yields a dense countable subset

{AT(sa 1)PrA. . .AT(s1)Prx|sa 1, . . . ,s12[0,1)\Q} ofK.

Theorem 5.3 then assures the existence of a set M ⇢ [0,1)with density 1 such that

sj2M,limsj!1T(s_j)P_sy=0 weakly for everyy 2K