OF OPERATORS ON SPACES OF CONTINUOUS FUNCTIONS
EDUARD YU. EMEL’YANOV and RADU ZAHAROPOL
We study the pointwise and norm convergence of Lotz-R¨abiger nets of operators defined on spaces of continuous functions. Our main result is a pointwise conver- gence theorem for such nets. The theorem is a natural extension of a pointwise mean ergodic theorem [12, Corollary 4.3.2] and a partial extension of [13, Theo- rem 1.1].
AMS 2010 Subject Classification:Primary 47A35; Secondary 37A30, 60J05, 60J25, 60J35.
Key words: operator net, transition probability, transition function, pointwise convergence, equicontinuity.
1. INTRODUCTION
Averages of operators appear in many fields of mathematics (for in- stance, in ergodic theory, probability theory, and operator theory), in theo- retical physics, and in various other areas in science. Generally, when dealing with a collection of operator averages, there is a natural way to structure the collection as a sequence, or more generally, as a net of operator averages, and, of course, when we consider a sequence or a net of operator averages we are interested in its convergence behaviour with respect to various kinds of convergence that make sense in the given setting. In an effort to extend the concept of net of operator averages, the first author and Erkursun introduced and studied the notion of Lotz-R¨abiger net in [4]; the notion has emerged from the papers of Lotz [7] and R¨abiger [8], and has been studied further in [3]. When proving convergence theorems for Lotz-R¨abiger nets, one usually extends and/or unifies results and approaches that deal with the convergence of nets of operator averages. Our goal in this paper is to study the pointwise and norm convergence of Lotz-R¨abiger nets of operators defined on spaces of continuous functions.
In this section, we review briefly some of the terminology, notation, and basic results used in the present work. Also in this section, we state the main
REV. ROUMAINE MATH. PURES APPL.,55(2010),1, 1–26
result of the paper, which is a pointwise convergence theorem for a certain type of Lotz-R¨abiger nets of operators on spaces of continuous functions.
1.1. Given a Banach space E, we denote by L(E) the Banach algebra of all continuous linear operators onE.
Let (X, ρ) be a locally compact separable metric space(an LCSM-space, in short). Examples of LCSM-spaces are: discrete countable metric spaces, open subsets ofRn, and compact metric spaces. Every LCSM-space is a union of countably many compact subsets (cf. [12, Proposition 1.1.3]).
In the present paper, we deal mainly with the Banach spaces C0(X)⊆Cb(X)⊆Bb(X) and M(X)∼=C0∗(X),
whereBb(X) is the space of all real-valued bounded Borel functions on (X, ρ) with respect to the sup-norm denoted by k · k∞, Cb(X) is the subspace of Bb(X) of all continuous functions, C0(X) is the subspace of Cb(X) of all functions vanishing at infinity, and M(X) stands for the Banach lattice of all real-valued signed measures on the Borel σ-algebra B(X). Note that if X is compact, then C0(X) =Cb(X); thus, in this case, we denote byC(X) any of the spaces C0(X) or Cb(X).
1.2. A family F of real-valued functions on X is called equicontinuous if for anyε >0 andx∈X there existsδ >0 such that
(1.1) ρ(x, y)< δ⇒ |f(x)−f(y)|< ε ∀f ∈ F.
It is easy to see that the above definition can be reformulated as follows:
The family F is equicontinuous if and only if for every x ∈ X and every convergent sequence (xn)n∈N of elements of X such that lim
n→∞xn=x and for every ε >0 there exists nε ∈Nsuch that |f(xn)−f(x)|< ε for everyf ∈ F and n∈N,n≥nε.
Every element of an equicontinuous family is a continuous function.
If Λ is a directed partially pre-ordered set (a directedPPO-set,in short), and if Γ = (fλ)λ∈Λis a net of real functions, we say that Γ is equicontinuous if the range{fλ|λ∈Λ}of Γ is equicontinuous. (By a preorder we mean a binary relation which is transitive and reflexive (see Schaefer’s monograph [10]).) In particular, a sequence of functions (fn)n∈N, fn : X → R for every n ∈ N is equicontinuous if the set {fn|n∈N} is equicontinuous.
We also need the following stronger type of equicontinuity: A family F of real-valued functions on X is called uniformly equicontinuous if for every ε >0 there exists δ >0 such that
(1.2) ρ(y, z)< δ⇒ |f(y)−f(z)|< ε ∀f ∈ F.
Every element of a uniformly equicontinuous family is a uniformly con- tinuous function.
As in the case of the usual (not necessarily uniform) equicontinuity, we say that a net (fλ)λ∈Λ of real-valued functions defined on X is uniformly equicontinuous if {fλ|λ ∈ Λ} is a uniformly equicontinuous set. In parti- cular, a sequence (fn)n∈N of real-valued functions defined on X is uniformly equicontinuous if {fn|n∈N} is a uniformly equicontinuous set.
The following elementary lemma is well-known, but we present a proof for the sake of completeness.
Lemma 1.1. Assume that (X, ρ) is a compact metric space. Then every equicontinuous family F ⊆C(X) is uniformly equicontinuous.
Proof. Here and throughout the paper, B(a, r) = BX(a, r) stands for the open ball of radius r centered at an element a of the metric space X under consideration.
LetF ⊆C(X) be equicontinuous and letε >0. Then, for every x∈X, there exists δ(x) >0 satisfying
(1.3) y∈B(x, δ(x))⇒ |f(x)−f(y)|< ε
3 ∀f ∈ F.
Since X is compact, there exists a finite family {xi}mi=1 ⊆X such that X = Sm
i=1
B xi,13δ(xi)
. Setδ:= min
1≤i≤m 1 3δ(xi).
Let u, z ∈ X be such that ρ(u, z) < δ. Then u ∈ B xk,13δ(xk) and z ∈ B(xp,13δ(xp)) for some k, p ≤ m. Assume that max{δ(xk), δ(xp)} = δ(xp) (the case max{δ(xk), δ(xp)}=δ(xk) can be treated in a similar manner). Then
ρ(xk, xp)≤ρ(xk, u) +ρ(u, z) +ρ(z, xp)< 1
3δ(xk)+δ+1
3δ(xp) ≤ (1.4)
≤ 1
3(δ(xk)+δ(xp)+δ(xp))≤δ(xp). Note that
ρ(u, xk)< 1
3δ(xk)< δ(xk), ρ(xp, z)< 1
3δ(xp)< δ(xp). (1.5)
An application of (1.3) to each of the inequalities in (1.4) and (1.5) gives
|f(u)−f(xk)|< ε
3, |f(xk)−f(xp)|< ε
3, |f(xp)−f(z)|< ε
3 ∀f ∈ F.
Consequently, |f(u)−f(z)|< ε for all f ∈ F.
We have therefore proved that for everyε∈R,ε >0, there existsδ∈R, δ >0 such that|f(u)−f(z)|< εfor everyf ∈ F whenever ρ(u, z)< δ.
1.3. Let Θ = (Sξ)ξ∈Ξ be a net of elements of L(Bb(X)). Then the net Θ is called C0(X)-equicontinuous if the range {Sξf : ξ ∈ Ξ} of Θ is equicontinuous for every f ∈ C0(X). The meaning of a C0(X)-uniformly equicontinuousnet is clear. Let S∈ L(Bb(X)). We say that the net Θ
(a)C0(X)-convergestoS if (Sξf)ξ∈Ξ converges toSf in the norm topo- logy of Bb(X) for everyf ∈C0(X);
(b)C0(X)-pointwise converges toS if
limξ∈ΞSξf(x) =Sf(x) ∀f ∈C0(X), x∈X.
1.4. A functionP :X× B(X) → [0,1] is called atransition probability (aTP, in short) if
(TP1) P(x0,·) is a probability measure for every x0 ∈X, and (TP2) P(·, A)∈Bb(X) for everyA∈ B(X).
One of the most trivial but rather useful TP is given in the next example.
Example 1.2. Let ν ∈ M(X) be a probability. Define a function P : X× B(X)→[0,1] by
(1.6) P(x, A) :=ν(A) ∀x∈X, A∈ B(X).
Then P is, obviously, a TP.
A minor extension of (1.6) is given in the following example.
Example 1.3. Let X =
∞
S
n=1
Xn be a pairwise disjoint union of Borel subsets and let (νn)∞n=1 be a sequence of probabilities, νn ∈ M(X) for every n∈N. Define a functionQ:X× B(X)→[0,1] by
(1.7) Q(x, A) :=
∞
X
n=1
χXn(x)·νn(A) ∀x∈X, A∈ B(X), where χXn denotes thecharacteristic function ofXn. Then Qis a TP.
1.5. LetP be a TP. There are two operators SP andTP associated with P. The operator SP :Bb(X)→Bb(X) is given by
(1.8) (SPf)(x) :=
Z
f(y)P(x,dy), ∀f ∈Bb(X), x∈X;
the operator TP :M(X)→ M(X) is given by (1.9) (TPµ)(A) :=
Z
A
P(x, A) dµ(x), ∀µ∈ M(X), A∈ B(X).
The operators SP and TP are both positive linear contractions, and sa- tisfy the condition hf, TPµi=hSPf, µi for allµ∈ M(X),f ∈Bb(X) (see [14, p. 53]); moreover,
SP(IX) =IX and kTPµk=kµk ∀0≤µ∈ M(X),
where IX =χX. Note also that
(1.10) (SPχA(x) =P(x, A)
for every x∈X and A∈ B(X).
In general, it may happen that SP(Cb(X)) 6⊆Cb(X) (see [12, Example 1.1.2]) andSP(C0(X))6⊆C0(X) even ifSP(Cb(X))⊆Cb(X) (use the TP from Example 1.2 in the case of a non-compact X).
1.6. Let S : Bb(X) → Bb(X). We say that S is generated by a TP if there exists a TP Qsuch thatS =SQ.
Using (1.10), we obtain that SQ1 = SQ2 implies Q1 = Q2. Thus, if a transition probability Q defines an operatorSQ :Bb(X)→Bb(X), then Qis unique in the sense that no other transition probability can be used to define SQin terms of (1.8). Note that the operatorTQ:M(X)→ M(X) defined by (1.9) has the property thatTQδx(A) =Q(x, A) for everyx∈XandA∈ B(X);
thus, TQ is uniquely determined by the TP Q and, therefore, by SQ. Here, and throughout the paper,δx stands for the Dirac measure concentrated atx;
that is,δx is the probability measure inM(X) such thatδx({x}) = 1,x∈X.
In the present paper, we study operator nets in L(Cb(X)). Except for Section 2, we restrict ourselves mostly to operator nets consisting of operators generated by TPs.
If S = SQ for a TP Q and S(Cb(X)) ⊆ Cb(X) then the pair (SQ, TQ) is called a Markov-Feller pair(anMF-pair, in short). The following result of Lasota and Yorke [6, Lemma 3.1] (cf. also [12, Theorem 1.2.4]) will be used in the proof of our main result (Theorem 1.7) in Section 3.
Proposition1.4 (The Lasota-Yorke Lemma). LetXbe an LCSM-space and let (S, T) be a Markov-Feller pair onX. Letφ:Cb(X)→Rbe a positive linear functional such that φ(Sf) = φ(f) for every f ∈ C0(X). Then the restriction µφ of φ to C0(X) has the property that T(µφ) =µφ provided that we think of µφ as an element of M(X).
1.7. Let (S, T) be an MF-pair. If X is an LCSM-space then, by the Rosenblatt theorem [9, p. 118] (see [12, Theorem 1.1.5] for the proof in the locally compact case), there exists a TP Qsuch thatS=SQ andT =TQ. In certain cases, it is easy to describe directly the TP Q.
LetQ:N× P(N)→[0,1] be a TP onN. ThenQis uniquely determined by the double sequence (αnk)∞k,n=1, where
(1.11) αnk =Q(n,{k}) ∀k, n∈N.
Hence, we may and do think of a TP Qon N as a double sequence (αnk)∞k,n=1 satisfying
(a)αnk ≥0 for allk, n∈N; (b)P∞
k=1αnk = 1 for all n∈N.
For everym∈N, denote byem the sequence em(i) =
1 i=m, 0 i6=m.
According to (1.8) and (1.9), any TPQ= (αnk)∞k,n=1onNdefines the operators SQ∈ L(`∞) and TQ ∈ L(`1) by
(1.12) (SQa)(j) =
∞
X
i=1
aiαji ∀a= (ai)∞i=1∈`∞, j∈N; in particular, αnk = (SQek)(n) for all k, n∈N.
(1.13) (TQb)(j) = (TQb)({j}) =
∞
X
n=1
Q(n,{j})bn=
∞
X
n=1
αnjbn ∀b= (bn)∞n=1∈`1; in particular, αnk = (TQen)(k) for all k, n∈N.
It is clear that if an operator S on Bb(N) satisfies S(en) = 0 for every n∈N then there is no TP that generatesS.
Example 1.5. Denote byPfin(N) the family of all finite subsets ofN. Let π ∈ Pfin(N),π6=∅, be given. Define thelocal averaging operatorAπ :`∞→`∞
by
(1.14) (Aπx)k= (Aπx)(k) :=
xk k6∈π, card(π)−1·P
i∈πxi k∈π
for allx= (xk)∞k=1 ∈`∞. ThenAπ is a positive contraction satisfyingAπI=I, whereI:=χN. The operatorAπ ∈ L(`∞) is generated by the TPQ= (αnk)∞k,n=1 on N, defined by
(1.15) αnk = (Aπek)(n) =
χ{k}(n) n6∈π, card(π)−1·P
i∈πχ{k}(i) n∈π.
1.8. LetEbe a Banach space, and letTbe a directed PPO-set. As usual, a net (Tα)α∈T of elements of L(E) is called anoperator net on E. Following [4], we say that (Tα)α∈T is a Lotz-R¨abiger net (an LR-net, in short) if the following two conditions are satisfied:
(LR1) (Tα)α∈T is uniformly bounded.
(LR2) lim
α∈T
kTαTβf −Tαfk = lim
α∈T
kTβTαf −Tαfk = 0 for every f ∈ E and β ∈T.
Examples of LR-nets can be found in the papers [3], [4], Lotz [7], R¨abiger [8], and in Krengel’s book [5, pp. 75–77]. Some examples will be discussed in this paper, as well.
When studying the strong (norm) convergence of LR-nets, we will use the following theorem obtained in Emel’yanov and Erkursun [4] (see also R¨abiger [8, Proposition 2.3] and Emel’yanov [3, Theorem 3.1]):
Theorem1.6 (Convergence Theorem for LR-nets). Let Θ = (Tλ)λ∈Λ be anLR-net on a Banach spaceX. Then the following conditions are equivalent:
(i) Θ converges strongly. In this case, the strong limit of the net Θ is a projection onto the fixed space Fix(Θ) of Θ ;
(ii)for every x∈X, the net (Tλx)λ∈Λ has a weak cluster point; (iii)
X = Fix(Θ)L S
λ∈Λ
(I−Tλ)X;
(iv) Fix(Θ) separates the fixed space Fix(Θ∗) of the adjoint net Θ∗ :=
(Tλ∗)λ∈Λ.
1.9. A sequence (sn)n∈N of elements of a directed PPO-set Λ is said to tend to infinity along Λ if, for every λ ∈ Λ, there exists nλ ∈ N such that λ ≤ sn for every n ∈ N, n ≥ nλ. The PPO-set Λ is said to be sequentially directed if there exists at least one sequence of elements of Λ that tends to infinity along Λ.
Note that a sequentially directed PPO-set is directed. The converse, how- ever, is not true. An example is NN, the set of all sequences of natural num- bers endowed with the pointwise ordering: (an)n∈N≤(bn)n∈Nif, by definition, an≤bnfor everyn∈N, (an)n∈N∈NN, (bn)n∈N∈NN. As expected, there exist many sequentially directed PPO-sets; for example, N, Rn, Rn+, n = 1,2, . . ., (Pfin(N),⊆) are all sequentially directed PPO-sets.
If Y is a nonempty set and (aλ)λ∈Λ is a net of elements of Y, then we say that (aλ)λ∈Λ is a sequentially directed net if Λ is a sequentially directed PPO-set.
The main result of this paper is the following theorem.
Theorem1.7 (Pointwise Convergence Theorem for LR-nets). Let X be an LCSM-space and let Θ = (St)t∈T be a C0(X)-equicontinuous sequentially directed Lotz-R¨abiger net of positive contractions onCb(X) generated by tran- sition probabilities. Then the net Θis C0(X)-pointwise convergent.
Theorem 1.7 is a natural extension of [12, Corollary 4.3.2] in the sense that Corollary 4.3.2 of [12] is a special case of Theorem 1.7. Indeed, let (S, T) be an MF-pair as in [12, Corollary 4.3.2]. Then, by the Rosenblatt theorem mentioned at 1.7,S is generated by a TP.
Now, in Theorem 1.7, let T=N, set ASn = n1
n−1
P
k=0
Sk and Qn= n1
n−1
P
k=0
Pk
for every n∈N, wherePk is the TP generatingSk,k∈N∪ {0}(note that the identity operator S0 is generated by the TP P0 defined by P0(x, A) = χA(x) for every x ∈ X and A ∈ B(X)). Clearly, (ASn)∞n=1 is a sequentially directed LR-net of positive contractions on Cb(X). Since in [12, Corollary 4.3.2] we assume that the sequence (Sn)∞n=0isC0(X)-equicontinuous, we obtain that the
LR-net (ASn)∞n=1 is C0(X)-equicontinuous, as well. Finally, since Qn is easily seen to be the transition probability that definesASn,n∈N, we conclude that [12, Corollary 4.3.2] is a special case of Theorem 1.7.
1.10. The paper is organized as follows: in Section 2, we discuss several results on the strong convergence of LR-nets onCb(X); in Section 3, we prove Theorem 1.7 and discuss an Ascoli-Arzel`a type result for separable metric spaces that we need in the proof; finally, in Section 4, we use Theorem 1.7 in order to study certain LR-nets defined by transition functions and obtain a result similar to [13, Theorem 1.1].
2. NORM CONVERGENCE OF LOTZ-R ¨ABIGER NETS ON Cb(X)
2.1. Here we present some convergence theorems forLR-nets onCb(X).
We begin with the following result.
Theorem2.1. Let X be a locally compact Hausdorff space, and letΘ = (St)t∈T be an LR-net on Cb(X) that satisfies the condition
(2.1) 0∈co(Θf) ∀f ∈C0(X).
Then lim
t∈T
kStfk∞= 0 for all f ∈C0(X).
Proof. Take an arbitrary f ∈C0(X). By condition (2.1), there exists a sequence (hn)∞n=1 of elements of co(Θf) satisfying
(2.2) lim
n→∞khnk∞= 0. Let hn be an element of the form
mn
P
k=1
αnkSµnk(f), where µnk are elements of T, and αnk are nonnegative reals such that
mn
P
l=1
αnl = 1 for every n∈N and k∈ {1,2, . . . , mn}. By condition (LR2) of the definition of an LR-net, we have
limt∈T
kSt◦Sµnkf −Stfk∞= 0 ∀n∈N,∀k= 1,2, . . . , mn, and hence
(2.3) lim
t∈T
kSthn−Stfk∞= 0 ∀n∈N.
Since the net Θ = (St)t∈T is uniformly bounded, we deduce from (2.2) and (2.3) that lim
t∈T
kStfk∞= 0.
Corollary 2.2. Let X be a locally compact Hausdorff space, and let Θ = (St)t∈T be an LR-net of positive operators onCb(X). Then the following conditions are equivalent:
(a)the net Θis C0(X)-convergent to zero;
(b) inf
t∈T
kStfk∞= 0 for all f ∈C0(X)+.
Proof. Since it is obvious that (a) implies (b), it is enough to prove (b) ⇒ (a). As usual, we denote sup
t∈T
kStkby M. By (LR1),M <∞.
Let ε > 0 and f ∈ C0(X). Then f = f+−f− in C0(X) and, by (b), there exist t1, t2 ∈T satisfying
(2.4) kSt1(f+)k∞≤ ε
2M and kSt2(f−)k∞≤ ε 2M. Using (LR2), we conclude that there exists t3 ∈Tsuch that (2.5) kSt3(f)−St3St1(f)k∞≤ ε
2 and kSt3(f)−St3St2(f)k∞≤ ε 2. Combining (2.4) with (2.5) we arrive at
kSt3(f)−St3St2(f+)k∞=kSt3(f)−St3St2(f+−f−)−St3St2(f−)k∞≤ (2.6)
≤ kSt3(f)−St3St2(f)k∞+kSt3St2(f−)k∞≤ ε
2 +MkSt2(f−)k∞≤ε, and
kSt3(f)−St3St1(−f−)k∞=kSt3(f)−St3St1(f+−f−)+St3St1(f+)k∞≤ (2.7)
≤ kSt3(f)−St3St1(f)k∞+kSt3St1(f+)k∞≤ ε
2+MkSt1(f+)k∞≤ε.
Since all operatorsSt are positive, the functionSt3St2(f+) is nonnegative and the function St3St1(−f−) if nonpositive. It follows from (2.6) and (2.7) that
−ε≤St3(f)(x)≤ε ∀x∈X,
and hencekSt3(f)k∞≤ε. Sinceε >0 andf ∈C0(X) were chosen arbitrarily, we have
t∈infT
kStfk∞= 0 ∀f ∈C0(X),
and hence 0∈co(Θf) for allf ∈C0(X). By applying Theorem 2.1 we obtain that lim
t→∞kStfk∞= 0 for all f ∈C0(X).
Corollary 2.3. Let X be a locally compact Hausdorff space, and let T be a Ces`aro bounded positive operator onCb(X) satisfying the condition
n→∞lim n−1Tnf = 0 ∀f ∈Cb(X).
Then either lim
n→∞kATnfk= 0for every f ∈C0(X)or inf
n∈N
kATnf0k>0for some f0 ∈C0(X)+. Here we denote byATn the sum 1nPn−1
k=0Tk.
Proof. It is enough to remark that under the above assumptions, the sequence (ATn)∞n=1 is an LR-net. Then we apply Corollary 2.2.
2.2. The following convergence result is a natural extension of the well known criterion for mean ergodicity of Markov operators onC(K) for a Haus- dorff compact space K (see, for example, Krengel [5, Proposition 5.1.1]).
Theorem 2.4 (Strong Convergence Theorem for an LR-net on a Com- pact Space). Let K be a compact Hausdorff space, and let Θ = (St)t∈T be an LR-net on the Banach space C(K). If Θ(f) is equicontinuous for every f ∈C(K), then Θ converges strongly.
Proof. Assume that Θ(f) is equicontinuous for everyf ∈C(K). Remark that Θ(f) is bounded for every f ∈ C(K) by (LR1). The set {Stf :t ∈ T} is pre-compact in the norm topology of the Banach space C(K) for every f ∈C(K), due to the Ascoli-Arzel`a theorem. Hence, the net (Stf)t∈Tpossesses a norm and, hence, a weak cluster point, sayg(f)∈C(K), for anyf ∈C(K).
Then, by Theorem 1.6, the LR-net Θ converges strongly.
If K is not compact, the situation becomes dramatically more compli- cated. In this case, we prove a partial version of Theorem 2.4 in the next section, namely Theorem 1.7, which is the main result of the present paper.
3. POINTWISE CONVERGENCE
3.1. Our goal in this section is to prove Theorem 1.7. We begin with an extension of the Ascoli-Arzel`a theorem for separable metric spaces (for the classical Ascoli-Arzel`a theorem see, for instance, Yosida’s book [11]).
Lemma 3.1. Let (Y, ρ) be a separable metric space. If (φn)∞n=1 is a uni- formly bounded and equicontinuous sequence of real-valued functions defined onY, then there exists a subsequence(φnk)∞k=1of(φn)∞n=1that converges point- wise on Y.
Proof. Take a countable set D={xj|j ∈N} ⊆Y which is dense in Y. The scalar sequence (φn(xj))∞n=1 possesses a convergent subsequence for every j ∈N. By the standard diagonal method, there exists a subsequence (φnk)∞k=1 of (φn)∞n=1 that converges pointwise on D.
We now prove that the subsequence (φnk)∞k=1 converges pointwise onY. To this end, let x ∈ Y and ε > 0. We will find kε ∈ N such that |φn
k0(x)− φnk00(x)|< εfor all k0, k00∈N satisfyingk0, k00≥kε.
SinceDis dense inY, there exists a sequence (yl)∞l=1of elements ofDsuch that lim
l→∞ρ(yl, x) = 0. The equicontinuity of (φnk)∞k=1 implies the existence of lε ∈Nsuch that |φnk(yl)−φnk(x)|< ε3 for everyl≥lε and everyk∈N.
Since ylε ∈ D, (φnk(ylε))∞k=1 is a convergent sequence. Therefore, there exists kε ∈N such that |φn
k0(ylε)−φnk00(ylε)|< ε3 for allk0 ∈N, k0 ≥kε and k00 ∈N,k00≥kε. We obtain
|φn
k0(x)−φnk00(x)| ≤ |φn
k0(x)−φnk0(ylε)|+|φn
k0(ylε)−φnk00(ylε)|+
+|φn
k00(ylε)−φnk00(x)|< ε 3+ ε
3+ ε 3 =ε, whenever k0 ≥kε and k00≥kε.
Therefore, we have proved that (φnk(x))∞k=1 is Cauchy and, hence, con- verges for allx∈Y.
3.2. The next lemma is probably known, but, for the sake of complete- ness, we provide a proof.
Lemma 3.2. Assume that (Y, ρ) is a compact metric space, let (ψn)∞n=1 be an equicontinuous sequence of real-valued functions defined on Y that con- verges pointwise on Y. Then (ψn)∞n=1 converges uniformly.
Proof. Note that the functions ψn, n ∈ N, are continuous because the sequence (ψn)∞n=1 is equicontinuous. SinceY is compact,ψn∈C(Y) for every n ∈N. Since C(Y) is a Banach space, it is obvious that the proposition will be completely proved if we show that (ψn)∞n=1 is a Cauchy sequence in the sup-norm.
To this end, letε >0. By Lemma 1.1, the sequence (ψn)∞n=1is uniformly equicontinuous. Therefore, there exists δ > 0 such that |ψn(x)−ψn(y)|< ε3 for every n ∈ N and x, y ∈ Y such that ρ(x, y) < δ. Since Y is compact, Y =
l
S
i=1
B(yi, δ) for somey1, y2, . . . , yl ∈Y.
Using the fact that (ψn)∞n=1is a pointwise convergent sequence, we obtain that (ψn(yi))∞n=1is a Cauchy sequence for eachi= 1,2, . . . , l. Therefore, there exists nε such that |ψn(yi)−ψm(yi)| < ε3 for all n ≥ nε, m ≥ nε, and i = 1,2, . . . , l.
Now, letx∈Y. Then there existsi∈ {1,2, . . . , l}such thatx∈B(yi, δ).
In view of the manner in whichδwas chosen, we obtain that|ψn(x)−ψn(yi)|<
ε
3 for everyn∈N. Accordingly, we obtain that
|ψn(x)−ψm(x)| ≤ |ψn(x)−ψn(yi)|+|ψn(yi)−ψm(yi)|+
(3.1)
+|ψm(yi)−ψm(x)|< ε
for all n ≥nε and m ≥nε. Since (3.1) holds for every x ∈ Y, the sequence (ψn)∞n=1 converges uniformly.
Lemma 3.3. (a) Let (X, d) be an LCSM-space, and let (Sn)∞n=1 be a sequence of elements of L(Cb(X)). Assume that sup
n∈N
kSnk < +∞ and that
(Sn)n∈N is C0(X)-equicontinuous. Then there exists a C0(X)-pointwise con- vergent subsequence (Snk)∞k=1 of (Sn)∞n=1.
(b) If (X, d) is a compact metric space then the subsequence (Snk)∞k=1, whose existence is stated at (a), converges strongly.
Proof. SetM = sup∞n=1kSnk. Then 0≤M <+∞.
(a) Note that it is enough to find a subsequence (Snk)∞k=1of (Sn)∞n=1 such that (Snkf)∞k=1 converges pointwise on X for everyf ∈C0(X),kfk∞≤1.
LetBC0(X) be the closed unit ball of C0(X), set E =BC0(X)×X, and let ρ:E×E →Rbe defined by
ρ((f, x),(g, y)) :=kf−gk∞+d(x, y) ∀f, g ∈BC0(X), x, y∈X.
Clearly, (E, ρ) is a metric space, and the topology defined by ρis the product topology on BC0(X)×X. Since bothBC0(X) and X are separable, (E, ρ) is a separable metric space. For every n∈N, letφn:E →Rbe defined by
φn((f, x)) :=Snf(x) ∀f ∈BC0(X), x∈X.
In view of the assertion made at the beginning of the proof, it is easy to see that a subsequence (Snk)k∈
N of (Sn)n∈Nis C0(X)-pointwise convergent if and only if the corresponding subsequence (φnk)k∈
N of (φn)n∈Nis pointwise convergent on E. Thus, in order to complete our proof, it is enough to show that the sequence (φn)∞n=1 is uniformly bounded and equicontinuous on E because, in this case, we will be able to apply Lemma 3.1 and to obtain the existence of a pointwise convergent subsequence, say (φnk)∞k=1, of (φn)∞n=1.
Since
|φn(f, x)|=|Snf(x)| ≤ kSnfk∞≤M· kfk∞≤M ∀f ∈BC0(X), x∈X, n∈N, the sequence (φn)∞n=1 is uniformly bounded.
We prove now that (φn)∞n=1 is equicontinuous. Thus, we have to prove that, for every convergent sequence ((fi, xi))∞i=1 of elements ofEand for every ε > 0, there exists iε ∈ N such that |φn((fi, xi))−φn((f, x))| < ε for every n∈ Nand every i≥iε, where (f, x) is the limit of ((fi, xi))∞i=1 in the metric space (E, ρ).
To this end, let ((fi, xi))∞i=1 be a convergent sequence in the metric space (E, ρ), and assume that (f, x) ∈ E is the limit of the sequence. Then, obvi- ously, lim
i→∞kfi −fk∞ = 0 and lim
i→∞d(xi, x) = 0. Let ε >0. Since (Sn)∞n=1 is C0(X)-equicontinuous and lim
i→∞d(xi, x) = 0, there existsi0ε∈Nsuch that
|Snf(xi)−Snf(x)|< ε
2 ∀n∈N, whenever i ≥ i0ε. Since lim
i→∞kfi−fk∞ = 0, there exists i00ε ∈ N such that kfi−fk∞< 2Mε for everyi≥i00ε. Setiε = max{i0ε, i00ε}.
Using the definition ofM and φn, we obtain
|φn((fi, xi))−φn((f, x))|=|Snfi(xi)−Snf(x)| ≤
≤ |Snfi(xi)−Snf(xi)|+|Snf(xi)−Snf(x)|< M · kfi−fk+ε 2 < ε for all i≥iε and n∈N.
(b) Assume that (X, d) is a compact metric space, and let (Snk)∞k=1be the subsequence of (Sn)∞n=1 obtained at (a). Then, for everyf ∈Cb(X) =C0(X), the sequence (Snkf)∞k=1 satisfies all the conditions of Lemma 3.2. Therefore, (Snkf)∞k=1 converges uniformly for every f ∈ Cb(X) and, hence, (Snk)∞k=1 converges strongly.
3.3. Let (X, d) be an LCSM-space. Recall that M(X) is the Banach lattice of all real-valued signed Borel measures on X, where the norm on M(X) is the usual total variation norm, and the order relation that defines the Banach lattice structure on M(X) is the standard one (µ ≤ ν if, by definition, µ(A) ≤ν(A) for every A ∈ B(X), where µ, ν ∈ M(X)). We will use the notation hf, µi forR
Xfdµ, where f ∈ Bb(X) and µ∈ M(X). Note that M(X)∼=C0∗(X), but, in general,M(X)6∼=Cb∗(X).
Let (µt)t∈Tbe a net of elements of M(X). As usual, we say that (µt)t∈T
converges in the weak* topology of M(X) if there existsµ∈ M(X) such that limt∈T
hf, µti=hf, µi ∀f ∈C0(X).
In this case, we say that µ is thew∗-limit of the net (µt)t∈T, and we use the notation µ=w∗-lim
t∈T
µt. We will need the following result.
Lemma 3.4. Let (µt)t∈T be a sequentially directed net in M(X), and µ∈ M(X). The following assertions are equivalent.
(a)The net (µt)t∈T converges in the weak* topology ofM(X) to µ.
(b) For every sequence (tn)∞n=1 of elements of T that tends to ∞ along T, the sequence (µtn)∞n=1 converges to µin the weak* topology of M(X).
Since the weak* topology is Hausdorff, the proof of Lemma 3.4 is obtained immediately from the following more general lemma, which is probably known.
However, for completeness, we provide full details.
Lemma3.5. Let (ft)t∈T be a sequentially directed net defined on a Haus- dorff topological space Y, and let y∈Y. The following assertions are equiva- lent.
(a)The net (ft)t∈T converges toy.
(b) For every sequence (tn)∞n=1 of elements of T that tends to ∞ along T, the sequence (ftn)∞n=1 converges to y.
Proof. The implication (a) ⇒(b) is obvious.
(b)⇒ (a): We will prove thatyis the limit of (ft)t∈T. Since (ft)t∈T is a sequentially directed net, we may and do pick a sequence (tn)∞n=1 of elements of Tthat tends to ∞along T.
Now assume that (ft)t∈T does not converge to y. Then there exists a neighborhood V ofy inY such that, for every t∈Tthere exists s∈T,s≥t, such that fs∈/V. In particular, for everyn∈N, there existssn∈T,sn≥tn, such that fsn ∈/ V. Clearly, the sequence (sn)∞n=1 tends to ∞ along T and (fsn)∞n=1 does not converge to y. We have obtained a contradiction since we assume that (b) holds true. Hence, the net (ft)t∈T converges to y.
3.4. Now letS :Cb(X)→Cb(X) be a positive contraction defined by a TP P, i.e., S =SP. Then (S, TP) is an MF-pair. As pointed out at Subsec- tion 1.6, the operator TP is uniquely determined by S. As in [12], we callTP the Markov operator defined by P.
Let (St)t∈T be a C0(X)-equicontinuous sequentially directed LR-net of positive contractions onCb(X) defined by a family (Pt)t∈Tof transition proba- bilities (the fact that (St)t∈T is defined by a family of transition probabilities (Pt)t∈T means, of course, that the operator St is defined by the transition probability Ptfor every t∈T).
For every t ∈ T, let Tt be the Markov operator defined by Pt. We call (Tt)t∈T the net of Markov operators defined by (St)t∈T (or by (Pt)t∈T). Set (3.2) H=
α∈TN
α= (tn)∞n=1is a sequence of elements ofT that tends to ∞ alongT, such that the sequence (Stn)∞n=1 isC0(X)-pointwise convergent
. Note that taking into consideration thatT is sequentially directed, and using (a) of Lemma 3.3, we obtain that the set Hdefined in (3.2) is nonempty.
Let α = (tn)∞n=1 ∈ H and f ∈ C0(X). In view of the definition of H, we obtain that lim
n→∞Stnf(x) exists for every x ∈ X. Thus it makes sense to define the function fα:X→Rby
(3.3) fα(x) := lim
n→∞Stnf(x) ∀x∈X.
Lemma3.6. Let (St)t∈T be aC0(X)-equicontinuous sequentially directed Lotz-R¨abiger net of positive contractions ofCb(X), defined by transition proba- bilities. Let α ∈ H and f ∈ C0(X). Then the function fα defined in (3.3) belongs to Cb(X). Moreover, fα ∈Fix((St)t∈T), which means that Stfα =fα
for every t∈T.
Proof. Let α = (tn)∞n=1 ∈ H and let f ∈ C0(X). Since St, t ∈ T, are contractions of Cb(X), the functionfα defined in (3.3) is bounded.
We will now prove that fα is continuous. To this end, let (xk)∞k=1 be a convergent sequence of elements of X, set x = lim
k→∞xk, and let ε > 0. Since
(St)t∈T is C0(X)-equicontinuous, there exists kε ∈ N such that |Stf(xk)− Stf(x)|< ε2 for every k≥kε and t∈T. Thus we obtain that
|fα(xk)−fα(x)|= lim
n→∞|Stnf(xk)−Stnf(x)| ≤ ε
2 < ε ∀k≥kε; accordingly, fα is continuous. Sincefα is bounded, fα∈Cb(X).
In order to complete the proof of the proposition, it remains to show that Stfα = fα for every t ∈ T. To this end, let t ∈ T. Since (Su)u∈T is a sequentially directed LR-net, and since (tn)n∈N diverges to ∞ along T, we obtain that lim
n→∞kStStnf −Stnfk∞= 0.
Keeping in mind that, in our setting,Ttis the Markov operator defined by the transition probability Pt (which, in turn, defines St), and using the Lebesgue dominated convergence theorem, we obtain
|Stfα(x)−fα(x)|=|hStfα−fα, δxi|=|hfα, Ttδx−δxi|=
= lim
n→∞|h(StStn−Stn)f, δxi| ≤ lim
n→∞kStStnf−Stnfk∞= 0 for every t∈Tand x∈X. Thusfα ∈Fix((St)t∈T).
3.5. If α = (tn)∞n=1 ∈ H and µ ∈ M(X), then we define the function µα :C0(X)→Rby
(3.4) µα(f) := lim
n→∞hStnf, µi= lim
n→∞hf, Ttnµi ∀f ∈C0(X).
The function µα is well defined since the sequence (hStnf, µi)∞n=1is convergent whenever f ∈ C0(X). It is also easy to see that µα is linear. If µ ≥ 0 then the functional µα is positive and, therefore, continuous. If µ∈ M(X) is not necessarily a positive element of M(X), then µ = µ+−µ−, and we obtain that µα is continuous in this case, as well. Thus, µα ∈ C0∗(X) ∼= M(X) for any µ ∈ M(X). In other words, the sequence (Ttnµ)∞n=1 converges to µα in the weak* topology of M(X).
Lemma3.7. Let (St)t∈T be aC0(X)-equicontinuous sequentially directed Lotz-R¨abiger net of positive contractions ofCb(X), defined by transition proba- bilities, and let (Tt)t∈T be the net of Markov operators defined by(St)t∈T. Then µα ∈Fix((Tt)t∈T), that is, Ttµα=µα for every t∈T.
Proof. Letα= (tn)∞n=1∈ H,µ∈ M(X), and t∈Tbe given. In order to prove that Ttµα=µα, we will use the Lasota-Yorke lemma (Proposition 1.4).
To this end, letLbe a Banach limit, and let Φαbe the positive linear functional on Cb(X) defined by
(3.5) Φα(f) =L((hStnf, µi)∞n=1) =L((hf, Ttnµi)∞n=1) ∀f ∈Cb(X).
Note that this formula makes sense since (hStnf, µi)∞n=1∈`∞.
Note also that, by (3.4), if f ∈C0(X) then the sequence (hStnf, µi)∞n=1 converges tohf, µαi, so Φα(f) =hf, µαifor everyf∈C0(X), i.e., Φα|C0(X)=µα. In order to apply the Lasota-Yorke lemma, we have to prove that Φα(Stf)
= Φα(f) for everyf ∈C0(X), that is, we have to prove that (3.6) L((hStnStf, µi)∞n=1) =L((hStnf, µi)∞n=1) ∀f ∈C0(X).
Since any Banach limit is equal to zero onc0, in order to prove (3.6) it is enough to show that lim
n→∞hStnStf −Stnf, µi = 0 for every f ∈ C0(X). Taking into consideration that (Su)u∈T is an LR-net and that the sequence (tn)∞n=1 tends to ∞ alongT, we obtain lim
n→∞kStnStf−Stnfk∞= 0 for every f ∈Cb(X), in particular, for all f ∈C0(X). Since
|hStnStf −Stnf, µi| ≤ kStnStf−Stnfk∞· kµk ∀f ∈C0(X), n∈N, it follows that the sequence (hStnStf −Stnf, µi)∞n=1converges to zero whenever f ∈C0(X), i.e., (hStnStf−Stnf, µi)∞n=1∈c0, and (3.6) holds.
The Lasota-Yorke lemma implies that Ttµα =µα.
3.6. We now discuss the last result needed to prove Theorem 1.7. The result is of interest in its own right.
Theorem 3.8 (Weak* Convergence Theorem for LR-nets). Let (St)t∈T
be a C0(X)-equicontinuous sequentially directed Lotz-R¨abiger net of positive contractions of Cb(X) defined by a family of transition probabilities, and let (Tt)t∈Tbe the net of Markov operators defined by(St)t∈T. Then the net(Ttµ)t∈T
converges in the weak* topology of M(X) for every µ∈ M(X).
Proof. Note that it is enough to prove the theorem for µ ∈ M+(X).
Thus, let µ∈ M+(X). By Lemma 3.4, in order to prove that (Ttµ)t∈T con- verges in the weak* topology of M(X), it is enough to prove that for every sequence α = (tn)∞n=1 of elements of T that tends to ∞ along T, the net (Ttnµ)∞n=1 weak* converges and its limit does not depend on the choice of the sequence α.
Since by (a) of Lemma 3.3 every sequence (tn)n∈N of elements ofTthat tends to ∞ alongT has a subsequence that belongs toH, and using Proposi- tion 4.2.1 of [12], we obtain that the theorem will be completely proved if we show that µα = µβ for every α ∈ H,α = (tn)n∈N, and β ∈ H, β = (rn)n∈N, whereµα = w*- lim
n→∞Ttnµand µβ = w*- lim
n→∞Trnµ(we used here the notation introduced at the beginning of Subsection 3.5).
To this end, let α= (tn)∞n=1 ∈ H and let β = (rn)∞n=1 ∈ H. Our goal is to prove that µα ≤ µβ, because the inequality µβ ≤ µα can be obtained by interchanging the roles of α and β, and, of course, the two inequalities imply that µα=µβ.
In order to prove that µα ≤ µβ, let L be a Banach limit, and let Φα : Cb(X)→ R be the positive linear functional onCb(X) defined in (3.5). It is easy to see that Φα|C0(X)=µα. Also, let ˜µα :Cb(X)→Rbe defined by
˜ µα(f) =
Z
fdµα ∀f ∈Cb(X).
Note that ˜µα is the standard extension of µα toCb(X) in the terminology of the subsection The Lasota-Yorke Lemma of [12, Section 1.2]. Letf ∈Cb(X), f ≥0. SinceXis an LCSM-space, there exists an increasing sequence (fn)∞n=1 of elements of C0(X) such that f(x) = lim
n→∞fn(x) for all x ∈ X. It follows from the monotone convergence theorem that
˜ µα(f) =
Z
fdµα= lim
n→∞
Z
fndµα= lim
n→∞hfn, µαi= lim
n→∞hfn,Φαi ≤Φα(f). Hence ˜µα(f)≤Φα(f) for everyf ∈Cb(X),f ≥0.
Clearly, the proof of the inequalityµα≤µβ will be completed if we show that hg, µαi ≤ hg, µβi for everyg∈C0(X),g≥0. To this end, letg∈C0(X), g≥0. By Lemma 3.7,Ttµα =µα for every t∈T. Thus, we obtain that (3.7) hSrng, µαi=hg, Trnµαi=hg, µαi ∀n∈N.
Since β∈ H, it follows that the sequence (Srng)∞n=1 converges pointwise on X.
Letgβ be the pointwise limit of (Srng)∞n=1. Clearly,gβ∈Cb(X), by Lemma 3.6.
Using the Lebesgue dominated convergence theorem and (3.7), we obtain
˜
µα(gβ) = lim
n→∞hSrng, µαi=hg, µαi. Using the fact that ˜µα≤Φα, we obtain hg, µαi= ˜µα(gβ)≤Φα(gβ) =L (hgβ, Ttnµi)∞n=1
.
Since, by Lemma 3.6, hgβ, Ttnµi = hStngβ, µi = hgβ, µi for every n∈ N, the terms of the sequence (hgβ, Ttnµi)∞n=1 are all equal tohgβ, µi; hence,
(3.8) hg, µαi ≤ hgβ, µi.
Since gβ is the pointwise limit of (Srng)∞n=1, µβ = w∗- lim
n→∞Trnµ, and the Lebesgue dominated convergence theorem, we obtain
(3.9) hgβ, µi= lim
n→∞hSrng, µi= lim
n→∞hg, Trnµi=hg, µβi.
In view of (3.8) and (3.9), we obtain hg, µαi ≤ hg, µβi for every g ∈ C0(X), g≥0. Thus, the inequalityµα≤µβ holds, and the proof is complete.
3.7. Using Theorem 3.8, we can prove Theorem 1.7 easily now. The details follow.
Proof of Theorem 1.7. Let (St)t∈T be a C0(X)-equicontinuous sequen- tially directed LR-net of positive contractions of Cb(X) defined by a family of
