Semigroups of Kernel Operators

Home , Kernel (set theory), Separating set

Moritz Gerlach

Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultätfür Mathematik und Wirtschaftswissenschaften der UniversitätUlm.

Vorgelegt von Moritz Gerlach aus Limburg an der Lahn im Jahr 2014. Tag der Pr¨ufung: 28. Mai 2014

Gutachter: Prof. Dr. Wolfgang Arendt Prof. Dr. G¨unther Palm Prof. Dr. Rainer Nagel

Amtierender Dekan: Prof. Dr. Dieter Rautenbach Contents

Introduction 1

1 Preliminaries 7

2 Weakly continuous operators on the space of measures 13 2.1 Norming dual pairs ...... 14 2.2 The lattice of transition kernels ...... 15 2.3 The sublattice of weakly continuous operators ...... 20

3 Mean ergodic theorems on norming dual pairs 25 3.1 Average schemes ...... 27 3.2 An ergodic theorem on norming dual pairs ...... 31 3.3 An ergodic theorem on the space of measures ...... 36 3.4 Counterexamples ...... 43

4 Kernel and Harris operators 47 4.1 Deﬁnition and basic properties ...... 48 4.2 Characterization by star–order continuity ...... 53 4.3 Triviality of the peripheral point spectrum ...... 61

5 Stability of semigroups 71 5.1 Stability of semigroups of Harris operators ...... 72 5.2 Doob’s theorem ...... 79 5.3 A Tauberian theorem for strong Feller semigroups ...... 81

Appendix 89

Bibliography 97

Index 103

Introduction

Semigroups of operators describe the evolution of linear systems and processes. As an illustrating example, let us consider the heat equation

 ∂ ∂2  u(t, x) = u(t, x) t ≥ 0, x ∈ R ∂t ∂2x  u(0, x) = u0(x) x ∈ R.

A solution u(t, x) of this equation describes the heat at a speciﬁc time t ≥ 0 at position x ∈ R. Hence, this equation models the diﬀusion of heat on the real line, starting with initial distribution u0 at time t = 0. It is well-known that this equation has a unique bounded solution for every continuous and bounded initial value u0. This solution is given by

1 Z ∞ (x − y)2 (T (t)u0)(x) = √ exp − u0(y) dy. 4πt −∞ 4t

The so deﬁned operators T (t), which map an initial value u0 to the corresponding solution at time t ≥ 0, are linear and satisfy the semigroup law T (t)T (s) = T (t+s) for all t, s ≥ 0. The latter means that the state of the system at time t + s equals the state at time t when the system is considered to be initialized with T (s)u0 instead of u0. The family of operators (T (t))t≥0 is called the Gaussian Semigroup. One important property of this semigroup is that each operator T (t) maps positive functions to positive ones. That is to say that if the heat is everywhere larger or equal than zero, it remains so for all time. Such semigroups are called positive. Intuitively, this is a typical property of semigroups describing some kind of diﬀusion. Moreover, the operators of the Gaussian Semigroup have an additional special property. Each operator T (t) is a kernel operator, i.e. it is of the form Z T (t)f = ht( · , y)f(y) dy

1 Introduction

for some function ht. In general, one cannot describe the operators of a semigroup by an explicit formula as in the example above. If, for instance, one is interested in the diffusion of heat on a plate, it is hard to impossible to calculate the single operators T (t) in one way or another unless a very specific geometry of the plate is given. Instead, there are often abstract arguments at hand to ensure that a semigroup consists of kernel operators although one does not know the kernel functions ht explicitly. For example, one can show that basically all differential operators generate a semigroup of kernel operators, see [12].

In this thesis we study several aspects of semigroups of kernel operators. First, we provide abstract characterizations of kernel operators which we use in particular to identify kernel operators on spaces of measures. Secondly, we study the asymptotic behavior of such semigroups, namely convergence of the semigroups in mean and convergence of the semigroups itself. It is well-known that kernel operators can be described in the following way. A regular and order continuous operator is a kernel operator if and only if it belongs to the band generated by the finite rank operators. If the latter is used as definition, this generalizes the notion of a kernel operator to arbitrary Banach lattices. On the other hand, kernel operators can be characterized by a continuity condition. Bukhvalov showed in [16] that an operator on Lp is a kernel operator if and only if it maps order bounded norm convergent sequences to almost everywhere convergent ones. We provide a generalization of Bukhvalov’s theorem for operators on Banach lattices and describe abstract kernel operators in terms of the so-called star–order continuity. This characterization is originally due to Grobler and van Eldik [35]. We give an alternative proof under different conditions on the spaces. The analysis of the asymptotic behavior of operator semigroups has a long history. Of particular interest is the characterization of stability, i.e. convergence to an equilibrium, by properties of the semigroup or its generator. Most results obtained in this area establish a connection between the asymptotic behavior and properties of the spectrum of the generator. Our approach is completely different and based on a technique developed by Greiner. He showed in [33] that a positive, contractive and irreducible strongly continuous semigroup on Lp converges to a projection onto its fixed space under the following two conditions: The peripheral point spectrum of its generator is {0} – in particular, the semigroup is assumed to admit a non-zero fixed point – and the semigroup contains two operators which are not disjoint. Due to a theorem of Axmann [14], the second condition holds automatically if the operators of the semigroup dominate kernel operators. Such operators are called Harris operators. Greiner proved in [33] that the first condition is satisfied for semigroups of kernel operators. Combining these result, Greiner

2 showed that a positive, contractive and irreducible strongly continuous semigroup on Lp converges to a projection onto its ﬁxed space if it consists of kernel operators and admits a non-zero ﬁxed point. This is an extraordinary theorem as it does not follow from one of the known spectral descriptions of stability. We generalize this theorem in several respects. First, we consider time-discrete semigroups in addition to strongly continuous ones and this not only on Lp but also on arbitrary Banach lattices. Secondly, we show that the peripheral point spectrum of the generator of a semigroup of Harris operators is trivial. As a consequence, we obtain a stability theorem for semigroups whose operators are merely assumed to dominate kernel operators.

Moreover, we present applications of Greiner’s technique to a diﬀerent but related class of positive semigroups. In the study of Markov processes one is interested in certain operators on spaces of measures which describe the evolution of probability distributions and are in general not kernel operators. In such a process, a state x is not transported deterministically to a certain state y. Instead, the new state is chosen by chance. The measure k(x, · ), the so-called transition kernel, is the distribution of states one is moved to from state x in one time step. In general, a Markov process carries a probability measure µ over to Z T µ = k(x, · ) dµ(x). Ω This deﬁnes a linear operator T on the space of measures which is also called kernel operator in the probabilistic literature. To distinguish an operator of this form from kernel operators as mentioned before, we call them weakly continuous operators as they are precisely those operators which are continuous with respect to the weak topology induced by the bounded measurable functions.

Here, we study probably for the first time order properties of weakly continuous operators. In doing so, we determine differences and similarities of weakly continuous operators and kernel operators. It turns out that the space of weakly continuous operators is a countably order complete sublattice of the bounded operators on the space of measures. However, in contrast to kernel operators which form a band, weakly continuous operators are not even an ideal. While it is nearly impossible to calculate the infimum of arbitrary regular operators in a practical way, the situation improves for weakly continuous operators. As for kernel operators, the computation of lattice operations of weakly continuous operators reduces to the corresponding lattice operations of their transition kernels. The infimum T1 ∧T2 of two weakly continuous operators is again weakly continuous and given by the infimum of their transition kernels k1 and k2 defined pointwise as the infimum of the measures k1(x, · ) and k2(x, · ).

3 Introduction

Therefore, it is now possible to impose conditions on the transition kernels to ensure that the corresponding operators are not disjoint. For example, it suﬃces to assume the measures k(x, · ) and k(y, · ) to be mutually absolutely continuous, which is a form of irreducibility. This enables us to apply Greiner’s theorem in this context. By this method, we obtain a purely analytic proof of a version of Doob’s theorem on stability of Markov processes and uniqueness of invariant measures. The same arguments also yield a version for time-discrete semigroups, i.e. for Markov chains.

A necessary condition for stability is convergence of the semigroup in mean, a property called mean ergodicity. This is in general strictly weaker than convergence of the semigroup itself and can typically be checked easily by the well-known mean ergodic theorem. This theorem asserts that a bounded semigroup is mean ergodic if and only if its fixed space separates the fixed space of the adjoint semigroup. While this characterization is suited e.g. for the Gaussian Semigroup on an Lp-space, it does not work well for semigroups on the space of measures. The reason is that the dual space of the measures with respect to the norm topology is too large to determine the fixed space of the adjoint semigroup. Moreover, it is not reasonable to expect strong convergence of the means of a Markov process with respect to the norm topology. Even though in some exceptional cases one obtains convergence of means or of the semigroup itself in the total variation norm, e.g. by Doob’s theorem, it is more natural to consider convergence in the weak topology σ(M (Ω),Cb(Ω)) induced by the bounded continuous functions. However, a characterization in the spirit of the classical mean ergodic theorem is still missing. This is provided in this thesis.

We work in the framework of norming dual pairs introduced in [46, 47] and consider simultaneously two semigroups which are related to each other via duality. From the point of view of applications to Markovian semigroups this is rather natural, as there are two semigroups dual to each other associated with a Markov process. The first acts on the space of bounded measurable functions on the state space Ω (or a subspace thereof such as Cb(Ω)) and corresponds to the Kolmogorov backward equation and the second acts on the space of bounded measures on Ω and corresponds to the Kolmogorov forward equation (or Fokker-Planck equation). We show that in general for semigroups and means on norming dual pairs, not all assertions corresponding to those of the classical mean ergodic theorem are equivalent. In particular, the separation of the fixed spaces is necessary but not sufficient for weak mean ergodicity. The situation improves when we restrict ourselves to the more special situation of Markovian semigroups on the norming dual pair (M (Ω),Cb(Ω)). Then an additional assumption, weaker than

4 the e-property, implies a characterization of weak mean ergodicity in analogy to the classical mean ergodic theorem.

While mean ergodicity is a priori weaker than convergence of the semigroup itself, it is a challenging task in analysis to find conditions under which convergence of means already implies convergence of the semigroup. Results of this type are called Tauberian theorems. We prove such a theorem for a certain class of Markovian semigroups on the space of measures. A bounded operator on the space of measures is said to have the strong Feller property if its adjoint maps bounded measurable functions to continuous ones. We prove that the square of a strong Feller operator is a kernel operator, i.e. it belongs to the band generated by the finite rank operators. Therefore, we are again in the position to apply Greiner’s theorem which yields the following Tauberian theorem: For a stochastically continuous Markovian semigroup of strong Feller operators, weak mean ergodicity – which is characterized by the separation of its fixed spaces as seen before – implies convergence of the semigroup to the mean ergodic projection in the total variation norm. In contrast to Doob’s theorem, we do not need to assume the semigroup to be irreducible which forced the fixed space to be one-dimensional. Instead, we obtain convergence of the semigroup even if its fixed space is of arbitrary high dimension.

The thesis is organized as follows. First we ﬁx some notations and recall the basics of the theories of semigroups and Banach lattices in Chapter 1. Our studies of the lattice structure of weakly continuous operators can be found in Chapter 2. We start with an introduction to the theory of norming dual pairs, the abstract framework for weakly continuous operators. Then we show in Section 2.2 that the transition kernels carry a lattice structure. Since the mapping of transition kernels to weakly continuous operators is positive and bijective with a positive inverse, this implies that weakly continuous operators are a vector lattice, too. The subsequent section is concerned with the comparison of the lattice structure of all regular operators and the weakly continuous ones. It is shown that the latter is a countably order complete sublattice which is not order complete and not an ideal. Chapter 3 contains our version of the mean ergodic theorem. After introducing the notation of an average scheme which serves as our generalized mean for time- discrete and time-continuous semigroups, we prove a version of the mean ergodic theorem on norming dual pairs in Section 3.2. Afterwards, in Section 3.3, we focus on more special average schemes on the norming dual pair (M (Ω),Cb(Ω)). In this situation, which covers the case of semigroups with the strong Feller or the e-property, we obtain that all assertions known from the classical mean ergodic theorem are equivalent. We conclude Chapter 3 with several counterexamples illustrating the optimality of the foregoing theorems.

5 Introduction

In the first section of Chapter 4 we introduce kernel and Harris operators and study their properties. After giving the precise definitions we show that on a diffuse Banach lattice kernel operators are disjoint from lattice isomorphisms while on atomic ones every operator is a kernel operator. The following section, Section 4.2, contains our characterization of kernel operators by star–order continuity. As a consequence, we obtain that for a semigroup of kernel operators, the space splits in at most countably many invariant bands where the restricted semigroup is irreducible. In the last section we study spectral properties of irreducible semigroups and prove that their peripheral point spectrum is trivial whenever the semigroup dominates a compact operator or a kernel operator. The final chapter contains our main results on stability of semigroups. In Section 5.1 we prove that every irreducible time-continuous semigroup of Harris operators with a non-trivial fixed space converges strongly to its mean ergodic projection. The same holds for a time-discrete semigroup under the additional assumption that it is expanding. In Section 5.2 we give our alternative proof of Doob’s theorem. Due to the lattice structure of the space of weakly continuous operators, this is now an easy consequence of Greiner’s theorem. In addition, we obtain a time-discrete version, i.e. a stability result for Markov chains. We conclude this chapter with our Tauberian theorem in Section 5.3 which states that for a strong Feller semigroup weak mean ergodicity is equivalent to stability. In the appendix, we provide the proof of Greiner’s zero-two law, which is the key to his before mentioned stability theorem. In a second part, we give a proof of Axmann’s theorem.

Acknowledgements

It is my great pleasure to express my gratefulness to everyone who supported me and my work over the last years. First of all, I warmly thank my advisor Wolfgang Arendt for his encouragement and acceptance into his research group. I am very grateful for his guidance and extremely competent assistance as well as for allowing me great latitude. I would like to thank all of my colleagues at the Institute of Applied Analysis, in particular Stephan Fackler, Jochen Gl¨uck and Robin Nittka for many enlightening discussions and especially Markus Kunze for the fruitful collaborations I enjoyed very much. I appreciate the great time I had at the Institute of Applied Analysis. I am also very grateful to my parents, who opened up so many possibilities to me, and to Nina for her support for many years. Lastly, I thank the graduate school Mathematical Analysis of Evolution, In- formation and Complexity for the ﬁnancial support in the years 2010 and 2011.

6 CHAPTER 1 Preliminaries

We give a short introduction to the theory of Banach lattices and positive operators. For further details, we refer to the monographs [55], [4] and [49]. We also introduce the concept of a semigroup and ﬁx some notation in this context. See [23] and [17] for an exposition of the general theory of strongly continuous semigroups and [13] for a special treatment of positive semigroups.

A non-empty set M with a relation ≤ is called partially ordered if the following conditions hold for every x, y, z ∈ M.

(a) x ≤ x,

(b) x ≤ y and y ≤ x implies x = y,

We also write x ≥ y for y ≤ x and x < y shorthand for x ≤ y and x 6= y. Let M be partially ordered and A ⊂ M. An element x ∈ M is called an upper (lower) bound of A if x ≥ y (x ≤ y) holds for all y ∈ M. If there exists an upper (lower) bound of A, then A is said to be bounded from above (below). We say that A is order bounded if A is both bounded from above and below. A minimum (maximum) of all upper (lower) bounds of A is called supremum (inﬁmum) of A. Such an element – if existent – is uniquely determined and denoted by sup A (inf A). A partially ordered set (M, ≤) is called a lattice if for every x, y ∈ M there exist sup{x, y} and inf{x, y}.

7 1. Preliminaries

Let E be a real vector space E endowed with a partial ordering. An element x ∈ E is called positive if x ≥ 0 and the set

E+ := {x ∈ E : x ≥ 0}

is called the positive cone of E. If in addition E is a lattice such that

x ≤ y ⇒ x + z ≤ y + z and αx ≤ αy

for all x, y, z ∈ E and α > 0, then E is a vector lattice (or Riesz space). In a vector lattice E, we deﬁne

x+ := x ∨ 0, x− := (−x)+ and |x| := x+ + x−.

If for every non-empty order bounded set A ⊂ E, sup A and inf A exist, then E is said to be order complete (or Dedekind complete). A vector lattice E endowed with a norm such that |x| ≤ |y| implies kxk ≤ kyk is called a normed vector lattice. If a normed vector lattice E is complete with respect to its norm, then E is called a Banach lattice. A subspace F of a vector lattice E is called a sublattice if it is closed under the lattice operations or, equivalently, if for each x ∈ F also |x| ∈ F . A subspace F with the additional property that |x| ≤ |y| for some y ∈ F implies that x ∈ F is called an ideal. Note that every ideal is a sublattice. If an ideal F ⊂ E is closed under suprema, i.e. sup A ∈ F for each set A ⊂ F whose supremum exists in E, then F called a band. In a normed vector lattice, every band is closed [49, Prop 1.2.3]. Let E be a vector lattice. Two elements x, y ∈ E are said to be disjoint if |x| ∧ |y| = 0. We deﬁne the disjoint complement of a set A ⊂ E as

A⊥ := {x ∈ E : |x| ∧ |y| = 0 for all y ∈ A}.

Then A⊥ is a band and, if E is order complete, A⊥⊥ is the band generated by A, i.e. the smallest band containing A. If a band B ⊂ E satisﬁes E = B ⊕ B⊥, then B is called a projection band. In an order complete vector lattice, every band is a projection band [49, Thm 1.2.9]. Before going any further, let us present some important examples. For every measure space (Ω, Σ, µ) and every 1 ≤ p ≤ ∞, the space of p-integrable functions Lp(Ω, Σ, µ) is a Banach lattice with respect to the ordering

f ≤ g :⇔ f(x) ≤ g(x) for µ-almost every x ∈ Ω.

If p < ∞ or the measure space (Ω, Σ, µ) is σ-ﬁnite, then Lp(Ω, Σ, µ) is order complete.

8 If Ω is a topological space, the bounded Borel measurable functions Bb(Ω) and the bounded continuous functions Cb(Ω), both endowed with the uniform norm k · k∞, form a Banach lattice with respect to the pointwise ordering. These spaces are in general not order complete. For a compact topological space K, Cb(K) = C(K) is order complete if and only if K is extremely disconnected (or Stonian), i.e. the closure of every open set is open [49, Prop 2.1.4]. Of particular importance in this thesis is the space of signed (and in particular ﬁnite) measures on a measurable space (Ω, Σ) endowed with the norm of total variation. This is also an order complete Banach lattice with respect to the setwise ordering µ ≤ ν :⇔ µ(A) ≤ ν(A) for all A ∈ Σ, in which the lattice operations are given by (µ ∨ ν)(B) = sup{µ(A) + ν(B \ A): A ⊂ B, A ∈ Σ} and (µ ∧ ν)(B) = inf{µ(A) + ν(B \ A): A ⊂ B, A ∈ Σ}, see [3, Sec 10.11]. Let E be a Banach lattice. A subset A ⊂ E is called downwards (upwards) directed if for every x, y ∈ A there exists z ∈ A such that x, y ≥ z (x, y ≤ z). We say that the norm on E is order continuous if inf{kxk : x ∈ A} = 0 for every downwards directed set A ⊂ E with inf A = 0. Every Banach lattice with an order continuous norm is order complete. Moreover, the norm on E is order continuous if and only if every closed ideal in E is a band [49, Sec 2.4]. The Banach lattice E is called an L-space if kx + yk = kxk + kyk for all x, y ∈ E+. If kx ∨ yk = max{kxk, kyk} for all x, y ∈ E+, then E is called an M-space. The norm of an L-space is always order continuous whereas the norm of an M-space in general is not. A vector x ∈ E+ of a vector lattice E is called a weak unit of E if E is the band generated by x. If even the generated ideal Ex equals E, then x is called a unit (or strong unit). A vector x ∈ E+ of a normed vector space E is said to be a quasi-interior point of E+ if the ideal generated by x is dense in E. Every quasi-interior point of E+ is a weak unit, while the converse holds if the norm is order continuous. Coming back to the foregoing examples, let us note that for every measure space (Ω, Σ, µ), the norm of Lp(Ω, Σ, µ) is order continuous for 1 ≤ p < ∞, where L1(Ω, Σ, µ) is even an L-space. The Banach lattice of signed measures on (Ω, Σ) ∞ is also an L-space and L , Bb and Cb are M-spaces. Let E and F be normed vector lattices. For linear operators T and S from E to F , we deﬁne T ≤ S :⇔ T x ≤ Sx for all x ∈ E+.

9 1. Preliminaries

Then L r(E,F ) := span{T : E → F linear : T ≥ 0} is a partially ordered vector space and its elements are called regular operators. If E is a Banach lattice, then every regular operator is bounded [49, Prop 1.3.5]. A linear operator T ≥ 0 is called positive. If T : E → F is bijective and positive with a positive inverse, then T is called a lattice isomorphism. A linear operator T : E → E is said to be irreducible if {0} and E are the only closed ideals left invariant by T . A linear projection P : E → E is called a band projection if 0 ≤ P ≤ I. In this case, B := PE is a projection band and I − P is the band projection onto B⊥. A linear operator T : E → F is called order bounded if it maps order bounded sets to order bounded ones. Clearly, every regular operator is order bounded. If F is order complete, then every order bounded operator is regular and L r(E,F ) is an order complete vector lattice, where

T +x = sup{T y : y ∈ E, 0 ≤ y ≤ x} T −x = − inf{T y : y ∈ E, 0 ≤ y ≤ x} |T |x = sup{|T y| : y ∈ E, |y| ≤ x} (T ∨ S)x = sup{T (x − y) + Sy : y ∈ E, 0 ≤ y ≤ x} (T ∧ S)x = inf{T (x − y) + Sy : y ∈ E, 0 ≤ y ≤ x}

r r for all x ∈ E+ and T,S ∈ L (E,F )[49, Thm 1.3.2]. We write L (E) shorthand for L r(E,E) and E∗ shorthand for L (E, R) = L r(E, R). Note that E is an L-space if and only if E∗ is an M-space and E is an M-space if and only if E∗ is an L-space [49, Prop 1.4.7]. A net (xα)α∈Λ in E is called order convergent to x if there exists a decreasing net (zα)α∈Λ with inf zα = 0 such that |xα − x| ≤ zα for every index α ∈ Λ; in r symbols o-lim xα = x. An operator T ∈ L (E,F ) is called order continuous if o-lim T xα = T x for every net (xα) ∈ E that order converges to x. If o-lim T xn = T x for every sequence (xn) ∈ E that order converges to x, then T is called r r ∗ ∗ countably order continuous. We write L (E,F )oc, L (E,F )coc, Eoc and Ecoc for the subspaces of order continuous and countably order continuous operators and functionals. If the norm on E is order continuous, then so is every regular operator from E to F . Since the theory of Banach lattices is inherently real, we assume all appearing spaces to be defined over the real numbers. However, in the context of spectral theory, we have to consider their complexifications. Let E be a Banach lattice. Then for each z = x + iy in the complexification EC := E ⊕ iE of E there exists

|z| := sup{cos(ϕ)x + sin(ϕ)y : 0 ≤ ϕ ≤ 2π}

10 and satisﬁes the following conditions. (a) |z| = 0 if and only if z = 0.

(b) |λz| = |λ||z| for all λ ∈ C and z ∈ EC.

If we endow EC with the norm kzkC := k|z|k, then EC is a Banach space which is called a complex Banach lattice. Every linear operator T : E → F uniquely extends to a C-linear operator TC : EC → FC deﬁned as TC(x + iy) = T x + iT y. If T is regular and F is order complete, then |TCz| ≤ |T ||z| for all z ∈ EC [49, Sec 2.2].

Let us now turn to the deﬁnition of a semigroup. A family S ⊂ L (X) of bounded linear operators on a Banach space X is called a semigroup if S contains the identity and ST ∈ S for all S, T ∈ S . If ST = TS for all S, T ∈ S , then S is called Abelian. We say that a semigroup S ⊂ L (X) is time-discrete if

n S = {T : n ∈ N0} for an operator T ∈ L (X). If a semigroup S is indexed with positive real numbers, i.e. S = (T (t))t≥0, such that T (t)T (s) = T (t + s) for all t, s ≥ 0, then S is called time-continuous. A time-continuous semigroup (T (t))t≥0 that satisfies limt→0+ T (t)x = x for all x ∈ X is said to be strongly continuous. A semigroup is called bounded if supT ∈S kT k < ∞ and contractive if supT ∈S kT k ≤ 1. For a strongly continuous semigroup S = (T (t))t≥0 on a Banach space X, the operator A: D(A) → X defined as 1 Ax := lim (T (h)x − x) h↓0 h on its domain D(A) := {x ∈ X : lim(T (h)x − x) exists} h↓0 is called the generator of S . This is a closed and densely defined operator that determines the semigroup S uniquely [23, Thm 1.4]. Let S ⊂ L (E) be a semigroup on a Banach lattice E. Then S is called positive if every operator T ∈ S is positive. We say that S irreducible if {0} and E are the only closed ideals left invariant under every T ∈ S . Note that n a time-discrete semigroup S = {T : n ∈ N0} is irreducible if and only if T is irreducible, whereas in general an irreducible semigroup does not contain an irreducible operator, see [33].

CHAPTER 2 Weakly continuous operators on the space of measures

In the following we study bounded operators on M (Ω), the space of all signed Borel measures on a Polish space Ω, that are continuous with respect to the weak topology induced by the bounded measurable functions Bb(Ω). Such operators are called weakly continuous and are characterized by the condition that their adjoints leave the space Bb(Ω) invariant, i.e. the operators act on the dual pair (M (Ω),Bb(Ω)). For this reason, we start with an introduction to the theory of norming dual pairs in Section 2.1 following Kunze [46, 47]. This is the abstract framework for the duality of M (Ω) and Bb(Ω) as well as for operators that are continuous with respect the norm topology and the induced weak one. In the subsequent sections we establish and study the lattice structure of L (M (Ω), σb), the space of all weakly continuous operators. First, in Section 2.2, we recall that each T ∈ L (M (Ω), σb) is of the form Z T µ = k(x, · ) dµ(x) Ω for a mapping k from Ω × B(Ω) to R, the so-called transition kernel. Then we show that these transition kernels are a lattice with respect to the natural pointwise ordering. Since the mapping of transition kernels to their induced operators is a lattice homomorphism into L (M (Ω)), the lattice structure carries

13 2. Weakly continuous operators on the space of measures

over to L (M (Ω), σb). Thus, for every T ∈ L (M (Ω), σb) there exists a least upper bound of {T, −T } inside L (M (Ω), σb). On the other hand, the modulus of each T ∈ L (M (Ω), σb) also exists in the order complete lattice L (M (Ω)) of all bounded operators. Hence the question appears naturally if the modulus of T in L (M (Ω)) coincides with the one in the subspace L (M (Ω), σ). In Section 2.3 we give a positive answer to this question by showing that L (M (Ω), σ) is actually a countably order complete sublattice of L (M (Ω)). All results from Sections 2.2 and 2.3 originate from [28], a joint work with Markus Kunze.

2.1 Norming dual pairs

We start by recalling well-known properties and examples of norming dual pairs, cf. [29, Sec 2], [46], [47]. Deﬁnition 2.1.1. A norming dual pair is a triple (X,Y, h · , · i) where X and Y are Banach spaces and h · , · i is a duality between X and Y such that

kxk = sup{|hx, yi| : y ∈ Y , kyk ≤ 1} and kyk = sup{|hx, yi| : x ∈ X, kxk ≤ 1}.

Identifying y with the linear functional x 7→ hx, yi, we see that Y is isometrically isomorphic to a norm closed subspace of X∗, the norm dual of X, which is norming for X. We brieﬂy say that (X,Y ) is a norming dual pair if the pairing h · , · i is clear from the context. Let us give some examples of norming dual pairs. If X is a Banach space with norm dual X∗, then (X,X∗) and thus, by symmetry, also (X∗,X) is a norming dual pair with respect to the canonical duality h · , · i∗. If (Ω, Σ) is a measurable space, we write Bb(Ω) for the space of all bounded measurable functions on (Ω, Σ) endowed with the supremum norm and M (Ω) for the space of all signed measures on (Ω, Σ) endowed with the total variation norm. Then (Bb(Ω), M (Ω)) is a norming dual pair with respect to the duality Z hf, µi := f dµ. Ω If Ω is a Polish space, i.e. it is a topological space which is metrizable through a complete and separable metric, and Σ is the Borel σ-algebra B(Ω), then also (Cb(Ω), M (Ω)) is a norming dual pair. For the easy proofs of these facts we refer to [47, Sec 2]. On a norming dual pair (X,Y ) we are interested in locally convex topologies which are consistent with the duality.

14 2.2. The lattice of transition kernels

Deﬁnition 2.1.2. Let (X,Y ) be a norming dual pair. We call a locally convex topology τ on X consistent if (X, τ)0 = Y , i.e. every τ-continuous linear functional ϕ on X is of the form ϕ(x) = hx, yi for some y ∈ Y . Of particular importance are the weak topologies σ(X,Y ) and σ(Y,X) associated with the dual pair. To simplify notation, we often write σ for the σ(X,Y ) topology on X and σ0 for the σ(Y,X) topology on Y . To indicate convergence with respect to σ and σ0, we write σ- lim and σ0- lim, respectively. If the norming dual pair under consideration is (M (Ω),Bb(Ω)) for a Polish space Ω, we denote the weak topologies by

0 0 σb := σ(M (Ω),Bb(Ω)) and σb := σ (Bb(Ω), M (Ω)). 0 For the norming dual pair (M (Ω),Cb(Ω)) we write σc and σc shorthand for 0 σ(M (Ω),Cb(Ω)) and σ (Cb(Ω), M (Ω)). If τ is a topology on X, we write L (X, τ) for the algebra of τ-continuous linear operators on X. Identifying Y with a closed subspace of X∗, we obtain the following characterization of σ-continuity, see [47, Prop 3.1]. Lemma 2.1.3. An operator T ∈ L (X) belongs to L (X, σ) if and only if its ∗ ∗ norm adjoint T leaves Y invariant. In that case, kT kX = kT |Y kY . 0 ∗ For a σ-continuous operator T ∈ L (X, σ) we write T for the σ-adjoint T |Y . It follows easily from Lemma 2.1.3 that L (X, σ) is a subalgebra of L (X) which is closed in the operator norm.

For the norming dual pair (Cb(Ω), M (Ω)) on a Polish space Ω also the strict topology β0 is important.

Definition 2.1.4. Let F0 be the space of all functions ϕ on Ω which vanish at infinity, i.e. given ε > 0 there exists a compact set K with |ϕ(x)| ≤ ε for all x ∈ Ω \ K. The strict topology β0 on Cb(Ω) is the locally convex topology generated by the set of seminorms {pϕ : ϕ ∈ F0} where pϕ(f) := kϕfk∞. 0 The strict topology is consistent with the duality, i.e. (Cb(Ω), β0) = M (Ω), see [38, Thm 7.6.3], and it coincides with the compact open topology on norm bounded subsets of Cb(Ω) [38, Thm 2.10.4]. Moreover, it is the Mackey topology of the dual pair (Cb(Ω), M (Ω)) [57, Thm 4.5, 5.8], i.e. it is the finest locally convex topology on Cb(Ω) which yields M (Ω) as a dual space. In particular, 0 L (Cb(Ω), σc) = L (Cb(Ω), β0), see [43, 21.4(6)].

2.2 The lattice of transition kernels

Let Ω be a Polish space endowed with its Borel σ-algebra B(Ω). In the following we consider the norming dual pair (M (Ω),Bb(Ω)) and prove that the space

15 2. Weakly continuous operators on the space of measures

L (M (Ω), σb) of weakly continuous operators is a lattice. To this end, we associate them in a lattice isomorphic way with their transition kernels and prove that the latter carry a lattice structure.

Deﬁnition 2.2.1. A transition kernel on Ω is a map k : Ω × B(Ω) → R with the following properties:

(a) A 7→ k(x, A) is a signed measure for every x ∈ Ω and

(b) x 7→ k(x, A) is a measurable function for every A ∈ Σ.

The total variation of the measure k(x, · ) is denoted by |k|(x, · ). The transition kernel k is called bounded if supx∈Ω|k|(x, Ω) < ∞. To each bounded transition kernel k, one may associate two mutually adjoint operators T ∈ L (M (Ω)) and S ∈ L (Bb(Ω)) in the following way. Lemma 2.2.2. Let k be a bounded transition kernel. By setting Z (T µ)(A) := k(x, A) dµ(x) (2.2.1) Ω

for µ ∈ M (Ω) and A ∈ B(Ω) and Z (Sf)(x) := f(y)k(x, dy) (2.2.2) Ω

for f ∈ Bb(Ω) and x ∈ Ω, we obtain operators T ∈ L (M (Ω)) and S ∈ L (Bb(Ω)) ∗ ∗ satisfying T |Bb(Ω) = S and S |M (Ω) = T . Moreover,

kT k = kSk = sup|k|(x, Ω). x∈Ω

Proof. Let us start with the operator S. We denote by

C := sup|k|(x, Ω) < ∞ x∈Ω

the bound of k. For every indicator function f = 1A, where A ∈ B(Ω), we have that Sf = k( · ,A) is a bounded measurable function by deﬁnition of k. Thus, by linearity, Sf ∈ Bb(Ω) for every measurable simple function f. Approximating an arbitrary f ∈ Bb(Ω) uniformly by a sequence of simple functions (fn) satisfying kfnk∞ ≤ kfk∞, we obtain that Sf is measurable as the pointwise limit of Sfn and Z |(Sf)(x)| ≤ |f(y)| · |k|(x, dy) ≤ kfk∞ · C Ω

16 2.2. The lattice of transition kernels

for all x ∈ Ω. This shows that S ∈ L (Bb(Ω)) with kSk ≤ C. For a finite partition Z ⊂ B(Ω) of Ω and x ∈ Ω we define X fx,Z := sign k(x, A) · 1A A∈Z and obtain that X kSk ≥ |(Sfx,Z )(x)| = |k(x, A)|. A∈Z Taking the supremum over all finite partitions Z of Ω and all x ∈ Ω implies that kSk ≥ C. Now we turn to the operator T . Let µ ∈ M (Ω) and let (An) ⊂ B(Ω) be a sequence of pairwise disjoint Borel sets. By definition of k

∞ Z Z X (T µ)(∪n∈NAn) = k(x, ∪n∈NAn) dµ(x) = k(x, An) dµ(x) Ω Ω n=1 for all x ∈ Ω. Since

N X N k(x, An) = |k(x, ∪n=1An)| ≤ C n=1 for all x ∈ Ω and N ∈ N and since the constant C is µ-integrable,

∞ ∞ ∞ Z X X Z X k(x, An) dµ(x) = k(x, An) dµ(x) = (T µ)(An) Ω n=1 n=1 Ω n=1 by the dominated convergence theorem. Thus, T µ is σ-additive and therefore a measure. It follows immediately that hT µ, 1Ai = hµ, S1Ai for every A ∈ B(Ω). Hence, by linearity, hT µ, fi = hµ, Sfi for every measurable simple function. Approx- imating f ∈ Bb(Ω) uniformly by a sequence (fn) ⊂ Bb(Ω) of simple functions ∗ shows that hT µ, fi = hµ, Sfi holds for all f ∈ Bb(Ω). Thus, T |Bb = S and ∗ S |M (Ω) = T . The identity kT k = kSk now follows from Lemma 2.1.3. Lemma 2.2.2 is well-known in the literature, see e.g. [45, Lem 2.2.2]. Combining it with Lemma 2.1.3 we see that the operator given by (2.2.1) is σb-continuous 0 and the operator given by (2.2.2) σb-continuous. The following Lemma shows that σb-continuity characterizes operators of this form. Lemma 2.2.3. For T ∈ L (M (Ω)) the following are equivalent: (i) There exists a bounded transition kernel k such that T is given by (2.2.1).

17 2. Weakly continuous operators on the space of measures

∗ (ii) The norm adjoint T of T leaves Bb(Ω) invariant.

(iii) The operator T is continuous in the σb = σ(M (Ω),Bb(Ω)) topology. Proof. Lemma 2.2.2 shows that (i) implies (ii) which is equivalent to (iii) by ∗ 0 Lemma 2.1.3. If (ii) holds, then S := T |Bb(Ω) is σb-continuous by Lemma 2.1.3 and hence given by (2.2.2) for a bounded transition kernel k as shown in [47, Prop 3.5]. Thus, T is of the form (2.2.1) by Lemma 2.2.2.

Likewise, for an operator S ∈ L (Bb(Ω)) it follows from Lemma 2.1.3 that 0 ∗ S is σb-continuous if and only if S leaves the space M (Ω) invariant. Thus, 0 ∗ S ∈ L (Bb(Ω), σb) if and only if S satisﬁes the equivalent conditions of Lemma ∗∗ 2.2.3 if and only if S = S |Bb(Ω) is given by (2.2.2) for a bounded transition kernel k.

Deﬁnition 2.2.4. An operator T ∈ L (M (Ω)) is called weakly continuous if it satisﬁes the equivalent conditions in Lemma 2.2.3. In this case, the transition kernel k from (i) of the lemma is called the associated transition kernel. Recall from Section 2.1 that we write L (M (Ω), σb) for the space of all weakly continuous operators.

We order the transition kernels on Ω pointwise, i.e. k1 ≤ k2 if and only if k1(x, A) ≤ k2(x, A) for all x ∈ Ω and A ∈ B(Ω). With respect to this ordering the transition kernels form a lattice by Proposition 2.2.5 below. Important for its proof is especially the fact that the lattice operations on Cb(Ω) are β0-continuous. This follows from [54, V 7.1] since in the strict topology the origin has a neighborhood base of solid sets.

Proposition 2.2.5. If k : Ω × B(Ω) → R is a transition kernel, then also |k|:Ω × B(Ω) → [0, ∞) is a transition kernel.

Proof. We have to show that |k|( · ,A) is measurable for all A ∈ B(Ω). Since Cb(Ω) is norming for M (Ω), we have that

kk(x, · )k = |k|(x, Ω) = sup |hf, k(x, · )i| (2.2.3) f∈Cb(Ω) kfk≤1

for every x ∈ Ω. This remains obviously true if we replace Ω with a closed subset F of Ω. Now we construct a countable set D ⊂ Cb(F ) independent of x such that (2.2.3) holds even if we take the supremum only over the set D. It then follows that x 7→ |k|(x, F ) is measurable for the arbitrarily chosen closed set F as a supremum of countably many measurable functions. Thus, the Dynkin system

A := {A ∈ B(Ω) : x 7→ |k|(x, A) is measurable}

18 2.2. The lattice of transition kernels contains all closed sets and hence equals B(Ω). So fix a closed set F ⊂ Ω. By [47, Thm 6.3] there exists a countable set M ⊂ Cb(F ) such that for all measures µ ∈ M (F ), µ 6= 0, there exists f ∈ M with hµ, fi= 6 0. We denote by S := spanQ M the linear span of M with rational coefficients. Now we show that the β0-closure of S, which is the same as the β0-closure of span M, is dense in Cb(F ). Assume that there exists g ∈ Cb(F ) which does not belong to the β0-closure of span M. Then, by the Hahn-Banach theorem, we find µ ∈ M (F ) such that hµ, fi = 0 for all f ∈ span M whereas hµ, gi= 6 0. Since µ vanishes in particular on the separating set M, it follows that µ = 0. This is a contradiction. Now we define D := {f ∧ 1 ∨ (−1): f ∈ S}. Since S is β0-dense in Cb(F ) and the lattice operations are β0-continuous, D is β0-dense in the closed unit ball of Cb(F ). Let x ∈ Ω and f ∈ Cb(F ), kfk ≤ 1. Given ε > 0 we find g ∈ D such that |hg − f, k(x, · )i| ≤ ε. Hence,

From this it follows that

|k|(x, F ) = sup|hf, k(x, · )i| f∈D as desired. This ﬁnishes the proof.

Now let T1,T2 ∈ L (M (Ω)) be weakly continuous operators with associated transition kernels k1, k2. Noting that h1A,Tjδxi = kj(x, A) for j = 1, 2, we see that T1 ≤ T2 as operators on M (Ω) if and only if k1 ≤ k2 as transition kernels. Thus the correspondence between a weakly continuous operator and its transition kernel is actually a lattice isomorphism. We thus obtain immediately from Proposition 2.2.5 the following result.

Theorem 2.2.6. The space L (M (Ω), σb) is a lattice in its natural ordering inherited from L (M (Ω)).

An operator T ∈ M (Ω) that is σc = σ(M (Ω),Cb(Ω))-continuous and thus acting on the norming dual pair (M (Ω),Cb(Ω)) is often called a Feller operator in the literature. Such operators are in particular weakly continuous in the sense 0 0 0 of Deﬁnition 2.2.4. Indeed, by Lemma 2.1.3, T is a σc = σ (Cb(Ω), M (Ω))- continuous operator on Cb(Ω) and thus given by (2.2.2) for a bounded transition

19 2. Weakly continuous operators on the space of measures

kernel k by [47, Prop 3.5]. The right hand side of (2.2.2) also deﬁnes a bounded 0 linear operator on Bb(Ω), which we still denote by T . Since Cb(Ω) separates 0∗ the points in M (Ω), T = T |M (Ω) is given by (2.2.1) and therefore is weakly continuous by Lemma 2.2.3. Now the question arises naturally whether also the Feller operators form a lattice. Equivalently, if the norm adjoint of a weakly continuous operator T with transition kernel k leaves the space Cb(Ω) invariant, does the same hold for the operator given by the kernel |k|? The following example shows that this is not the case. Example 2.2.7. We consider the set Ω = (−N)∪N∪{∞}, where the neighborhoods of the extra point ∞ are exactly the sets which contain a set of the form {∞} ∪ {n, n + 1,... } ∪ {−n, −(n + 1),...} for some n ∈ N, whereas all other points are isolated. Note that Ω is homeomorphic with the space {0, ±n−1 : n ∈ N} endowed with the topology inherited from R. Thus Ω is Polish. We also note that a bounded function f : Ω → R is continuous if and only if f(n) → f(∞) and also f(−n) → f(∞) as n → ∞. Now we deﬁne the transition kernel ( δ − δ n ∈ N k(n, · ) := n n+1 0 n ∈ (−N) ∪ {∞}.

Then ( δ + δ n ∈ N |k|(n, · ) = n n+1 0 n ∈ (−N) ∪ {∞}.

Let T,U ∈ L (M (Ω), σb) denote the operators associated with k and |k|, respec- ∗ ∗ tively. Then T Cb(Ω) ⊂ Cb(Ω). However, U maps the continuous function 1Ω to the function 21N which is not continuous. This shows that the σ(M (Ω),Cb(Ω))- continuous operators do not form a sublattice of the σ(M (Ω),Bb(Ω))-continuous operators. Moreover, there exists no modulus of T in the σ(M (Ω),Cb(Ω))- continuous operators. Indeed, if S was such a modulus, then the transition kernel ∗ of S has to coincide with |k| on N ∪ (−N). In particular S 1Ω(n) = 21N(n) for ∗ n =6 ∞. But this shows that S 1Ω cannot be continuous. This is a contradiction.

2.3 The sublattice of weakly continuous operators

We still consider the norming dual pair (M (Ω),Bb(Ω)) for a Polish space Ω endowed with its Borel σ-algebra. We have seen in Theorem 2.2.6 that the space L (M (Ω), σb) of weakly continuous operators is a lattice, i.e. for each T ∈ L (M (Ω), σb) there exists a least upper bound of {T, −T } inside L (M (Ω), σb), which is called the modulus of T .

20 2.3. The sublattice of weakly continuous operators

Since M (Ω) is an L-space, every bounded linear operator on M (Ω) is regular and L (M (Ω)) is a Banach lattice with respect to the natural ordering, see [55, Thm IV 1.5]. Thus, every weakly continuous operator has a modulus in L (M (Ω)). A natural question is whether this modulus can be diﬀerent from the modulus in the space of all weakly continuous operators. This is not the case. In the following we show that the weakly continuous operators form a countably order complete sublattice of L (M (Ω)). In addition, we provide examples showing that L (M (Ω), σb) is neither an ideal nor order complete.

In order to show that L (M (Ω), σb) is a sublattice of L (M (Ω)), we show that + for T ∈ L (M (Ω), σb) the positive part T , taken in the vector lattice L (M (Ω)), is again weakly continuous. Moreover, if k is the transition kernel associated to + T , then T is associated to the transition kernel k+ = (|k| − k)/2. We recall that the positive part within L (M (Ω)) of a weakly continuous operator T with associated transition kernel k is given by Z T +µ = sup T ν = sup k(x, · ) dν(x) 0≤ν≤µ 0≤ν≤µ Ω Z (2.3.1) = sup g(x)k(x, · ) dµ(x) g∈Bb(Ω) Ω 0≤g≤1 for every positive measure µ, see [49, Thm 1.3.2].

Lemma 2.3.1. Let k be a transition kernel, α > 0 and let U = {Bn : n ∈ N} be a countable basis of the topology on Ω that is closed under ﬁnite unions. Then [ {k+( · , Ω) > α} = {k( · ,Bn) > α}. n∈N

Proof. If k(x, Bn) > α for some x ∈ Ω and n ∈ N, then clearly k+(x, Ω) ≥ k+(x, Bn) ≥ k(x, Bn) > α. This shows the inclusion “⊃”. Conversely, let x ∈ Ω with k+(x, Ω) > α be given. We consider the Hahn decomposition Ω = Ω+ ∪ Ω− of the measure k(x, · ). By assumption k(x, Ω+) = k+(x, Ω) > α. Since the measure k(x, · ) is regular, there exists an open superset U ⊃ Ω+ with k(x, U) > α. Since U is closed under ﬁnite unions, using the regularity of k(x, · ) again, we ﬁnd a base set Bn ∈ U with p(x, Bn) > α.

Lemma 2.3.2. Let T ∈ L (M (Ω), σb) with associated transition kernel k. Let α > 0 and A ∈ B(Ω) such that α1A < k+( · , Ω). Then

+ (T µA)(Ω) ≥ αµ(A) where µA denotes the measure µ(A ∩ · ).

21 2. Weakly continuous operators on the space of measures

Proof. Let (Bn)n∈N be a countable basis of the topology on Ω that is closed under ﬁnite unions. We deﬁne En := A ∩ {k( · ,Bn) > α} for n ∈ N. Then Lemma 2.3.1 yields that [ A = A ∩ {k+( · , Ω) > α} = En. n∈N

Deﬁning Ω1 := E1 and Ωn := En \ (∪k 1 we obtain a decomposition of A in disjoint sets. Fix ε > 0. By the regularity of µ we ﬁnd an index N ∈ N with [ ε µ Ω ≥ µ(A) − . n α n≤N

We now reﬁne the sets B1,...,BN further. We ﬁnd disjoint Borel sets B˜1,..., B˜M such that (i) given m ≤ M and n ≤ N the set B˜m is either contained in Bn or disjoint from Bn and (ii) we have [ [ B˜m = Bn. m≤M n≤N

We let N(m) := {n ≤ N : B˜m ⊂ Bn} so that Bn is the disjoint union of those B˜m where n ∈ N(m). By choosing g in (2.3.1) as the characteristic of the set ∪n∈N(m)Ωn, we ﬁnd that M M Z + X + X (T µA)(Ω) ≥ (T µA)(B˜m) ≥ k(x, B˜m) dµ(x). S m=1 m=1 n∈N(m) Ωn

Since the sets Ωn as well as the sets B˜m are disjoint, we have that

M M X Z X X Z k(x, B˜m) dµ(x) = k(x, B˜m) dµ(x) S Ω Ω m=1 n∈N(m) n m=1 n∈N(m) n N X X Z = k(x, B˜m) dµ(x) n=1 m≤M Ωn n∈N(m) N X Z = k(x, Bn) dµ(x). n=1 Ωn

As k( · ,Bn) > α on Ωn, we conclude that

N X Z [ k(x, Bn) dµ(x) > αµ Ωn ≥ αµ(A) − ε n=1 Ωn n≤N which completes the proof.

22 2.3. The sublattice of weakly continuous operators

Theorem 2.3.3. The L (M (Ω), σb) is a sublattice of L (M (Ω)).

Proof. Let T ∈ L (M (Ω), σb) with associated transition kernel k. We denote by S the weakly continuous operator with transition kernel k+ and we prove that T + = S. To that end, let µ > 0. Since it follows easily from (2.3.1) that T +µ ≤ Sµ, it suﬃces to show that (T +µ)(Ω) = (Sµ)(Ω). Let ε > 0 and

M X 1 f = αj Aj j=1 be a simple function with coeﬃcients αj > 0 and pairwise disjoint sets Aj ∈ B(Ω) such that f(x) < k+(x, Ω) for all x ∈ Ω and Z k+(x, Ω) − f(x) dµ(x) < ε. Ω Lemma 2.3.2 yields that

M M + X + X (T µ)(Ω) ≥ (T µAj )(Ω) ≥ αjµ(Aj) j=1 j=1 Z Z = f(x) dµ(x) ≥ k+(x, Ω) dµ(x) − ε Ω Ω = (Sµ)(Ω) − ε.

+ + Hence T µ = Sµ and thus, since µ was arbitrary, T = S ∈ L (M (Ω), σb). In contrast to kernel operators the weakly continuous operators are not a band in L (M (Ω)). The following example shows that they are not even an ideal. Example 2.3.4. Let Ω be a Polish space that admits atomless measures, e.g. Ω = R. Let P : M (Ω) → M (Ω) denote the band projection onto the band of ∗ ∗ atomless measures and deﬁne ϕ := P 1. Then 0 < ϕ ≤ 1 and ϕ ∈ M (Ω) \ Bb(Ω) since hϕ, δxi = 0 for all x ∈ Ω. For a measure µ > 0 consider the positive rank one operator T := ϕ ⊗ µ on M (Ω). Then T ≤ 1 ⊗ µ ∈ L (M (Ω), σb) but ∗ T 1 = µ(Ω)ϕ 6∈ Bb(Ω) and hence T 6∈ L (M (Ω), σb). We conclude this section with an investigation of order completeness of the sublattice L (M (Ω), σb). We prove that this space is countably order complete but not order complete. Let us start with a well-known lemma.

Lemma 2.3.5. Let (µn) be an increasing sequence of positive measures on B(Ω) and ν ∈ M (Ω) such that µn ≤ ν for all n ∈ N. Then µ := sup µn is given by µ(A) = sup µn(A) for all A ∈ B(Ω).

23 2. Weakly continuous operators on the space of measures

Proof. Let fn denote the density of µn with respect to ν and deﬁne f := sup fn. Then Z Z f dν = sup fn dν = sup µn(A) A n∈N A n∈N for all A ∈ B(Ω) by the monotone convergence theorem. Therefore, the mapping A 7→ sup µn(A) deﬁnes a measure on B(Ω) and thus (sup µn)(A) = sup µn(A) for all A ∈ B(Ω).

Theorem 2.3.6. Let (Tn) ⊂ L (M (Ω), σb) be a sequence of weakly continuous operators that is order bounded by an element of L (M (Ω)). Then sup Tn exists in L (M (Ω)) and is weakly continuous.

Proof. Since M (Ω) is order complete, S := sup Tn exists in L (M (Ω)). It remains to show that S ∈ L (M (Ω), σb). Note that by Theorem 2.3.3, T1 ∨ · · · ∨ Tn is again weakly continuous. Thus, replacing Tn by T1 ∨ · · · ∨ Tn − T1 and S by S − T1, we may assume that (Tn) is increasing and Tn ≥ 0 for all n ∈ N. We denote by kn the transition kernel associated with Tn. For x ∈ Ω and A ∈ B(Ω) we deﬁne

k(x, A) := sup kn(x, A) ≤ (Sδx)(A) ≤ kSk. n∈N Then k( · ,A) is measurable for all A ∈ B(Ω) and k(x, · ) is a measure by Lemma 2.3.5. Hence, k is a bounded transition kernel. Since (Tn) is increasing, we have for all µ ∈ M (Ω)+ that Z Z Sµ = (sup Tn)µ = sup(Tnµ) = sup kn(x, · ) dµ(x) = k(x, · ) dµ(x), n∈N Ω Ω where the last identity follows from Lemma 2.3.5 and the monotone convergence theorem.

The following example shows that L (M (Ω), σb) is not order complete, i.e. not every order bounded set has a supremum. Example 2.3.7. Let Ω be a Polish space such that there exists an unmeasurable set E ⊂ Ω, e.g. Ω = R. Let µ ∈ M (Ω) be a probability measure. For each x ∈ E we consider the weakly continuous rank one operator Tx := 1{x} ⊗ µ. Then the set T := {Tx : x ∈ E} is dominated by 1 ⊗ µ. Let us assume that S := supx∈E Tx exists in L (M (Ω), σb). Fix x ∈ Ω \ E and denote by P the band ⊥ ∗ projection onto {δx} . Since P f = f · 1Ω\{x} ∈ Bb(Ω) for all f ∈ Bb(Ω), P is σb-continuous by Lemma 2.1.3. Thus, SP ∈ L (M (Ω), σb) is an upper bound of T and hence (Sδx)(Ω) ≤ (SP δx)(Ω) = 0. On the other hand, for y ∈ E we have ∗ (Sδy)(Ω) ≥ (Tyδy)(Ω) = 1. Therefore, S 1 is not a measurable function. This contradicts our assumption that S ∈ L (M (Ω), σ).

24 CHAPTER 3 Mean ergodic theorems on norming dual pairs

A bounded operator T ∈ L (X) on a Banach space X is called mean ergodic if 1 Pn−1 j limn→∞ Anx exists for every x ∈ X, where An := n j=1 T denote the Ces`aro averages of the operator T . The classical mean ergodic theorem (see [44, §2.1 Theorems 1.1 and 1.3] or [21, Thm 8.20]) characterizes mean ergodic operators as follows.

Theorem 3.0.8. Let X be a Banach space and T ∈ L (X) be power-bounded, i.e. n supn∈NkT k < ∞. The following assertions are equivalent.

(i) The operator T is mean ergodic.

∗ (ii) The sequence (Anx) has a σ(X,X )-cluster point for all x ∈ X.

(iii) ﬁx(T ) separates ﬁx(T ∗), i.e. for all x∗ ∈ ker(I − T ∗), x∗ 6= 0, there exists x ∈ ker(I − T ) such that hx, x∗i= 6 0.

(iv) X = ﬁx(T ) ⊕ rgk · k(I − T ).

For a family S of linear operators on a vector space X let us denote by \ ﬁx(S ) := ker(I − S) S∈S

25 3. Mean ergodic theorems on norming dual pairs

its ﬁxed space and by

rg(I − S ) := {x − Sx : x ∈ X, S ∈ S }

the range of I − S . Likewise, a strongly continuous semigroup T = (T (t))t≥0 ⊂ L (X) is called mean ergodic if limt→∞ Atx exists for each x ∈ X, where the operators At ∈ L (X), given by 1 Z t Atx := T (s)x ds t 0 for x ∈ X and t > 0, are called the Ces`aro averages of the semigroup T . For a bounded strongly continuous semigroup T one has the following characterization of mean ergodicity in analogy to Theorem 3.0.8, see Lemma V.4.4. and Theorem V.4.5 of [23].

Theorem 3.0.9. Let T = (T (t))t≥0 ⊂ L (X) be a bounded strongly continuous semigroup on a Banach space X. The following assertions are equivalent:

(i) The semigroup T is mean ergodic.

(ii) For every x ∈ X there exists an increasing sequence (rn) ⊂ [0, ∞) with ∗ lim rn = ∞ such that (Arn x) has a σ(X,X )-cluster point for all x ∈ X. (iii) fix(T ) separates fix(T ∗), i.e. for all 0 6= x∗ ∈ fix(T ∗) there exists x ∈ fix(T ) such that hx, x∗i= 6 0. (iv) X = fix(T ) ⊕ rgk · k(I − S ). There are countless extensions of these classical mean ergodic theorems to more general situations. These include weakening the assumption of boundedness, considering more general semigroups and means other than the Cesàroaverages, and replacing the Banach space X with a locally convex space (X, τ), see e.g. [20, 50, 53]. An overview of these results and further references can be found in [44]. Mean ergodic theorems for semigroups on special classes of locally convex spaces are addressed in [1, 2]. Unfortunately, it seems that one cannot treat convergence of means of Markov processes in the weak topology σ(M (Ω),Cb(Ω)) with a mean ergodic theorem on locally convex spaces (X, τ). The reason for this is that the known results require the means to be equicontinuous with respect to τ, see [1, 2, 20, 53]. If τ is the weak topology σ(M (Ω),Cb(Ω)), equicontinuity seems a rather strong assumption which is not satisfied in interesting examples. The literature on weak mean ergodicity of Markovian semigroups is rather extensive, let us mention [42, 59, 63, 64]. In the following we characterize convergence of averages on norming dual pairs in the spirit of Theorem 3.0.8 and 3.0.9.

26 3.1. Average schemes

In doing so, we allow for general (in particular also non-commutative) semigroups and means – even though our main interest lies in Cesàroaverages of time-discrete or time-continuous semigroups – and study convergence of the means and its adjoints in the weak topologies induced by the dual pair. In our first main result (Theorem 3.2.4), we show that in this general situation the statements corresponding to (i) and (ii) in Theorem 3.0.9 are equivalent and imply the statements corresponding to (iii) and (iv). We also provide counterexamples to show that in general (the statements corresponding to) (iii) does not imply (iv) and neither (iii) nor (iv) imply (i) and (ii). Afterwards, we focus on the more special situation of Markovian semigroups on the norming dual pair (M (Ω),Cb(Ω)). Under a condition weaker than the e-property, which plays an important role in [42, 59], we prove in Theorem 3.3.9 that the statements corresponding to (i) – (iv) in Theorem 3.0.9 are all equivalent for the semigroup on M (Ω). Moreover, if the semigroup on M (Ω) is mean ergodic with respect to σ(M (Ω),Cb(Ω)), then also the semigroup on Cb(Ω) is mean ergodic even with respect to the strict topology. Considering semigroups on (M (Ω),Cb(Ω)) rather than on the single Banach space M (Ω) makes our assumption natural, in fact, it is necessary for the convergence we obtain. We start with introducing the notion of an “average scheme” which will act as our means. Then we take up our main line of study. First, we analyze convergence of average schemes on general norming dual pairs in Section 3.2, then the convergence of average schemes on (M (Ω),Cb(Ω)) under additional assumptions in Section 3.3. The final Section 3.4 contains our Counterexamples. All results from this chapter originate from a joint work with Markus Kunze and are published in our article [29].

3.1 Average schemes

Recall from Section 2.1 that we write σ shorthand the σ(X,Y ) topology on X and σ0 for the σ(Y,X) topology on Y . By Lemma 2.1.3 an operator T ∈ L (X) is σ-continuous if and only if its norm adjoint T ∗ leaves Y invariant. The set of σ-continuous operators on X are denoted by L (X, σ). For a family S ⊂ L (X) and x ∈ X we deﬁne

n n X X co(S x) := akSkx : ak ≥ 0, ak = 1, Sk ∈ S , n ∈ N , k=1 k=1 the convex hull of the orbit of x under S . Inspired by [20] we make the following deﬁnition.

27 3. Mean ergodic theorems on norming dual pairs

Deﬁnition 3.1.1. Let (X,Y ) be a norming dual pair. An average scheme on (X,Y ) is a pair (S , A ), where S ⊂ L (X, σ) is a semigroup with adjoint 0 0 S := {S : S ∈ S } and A = (Aα)α∈Λ ⊂ L (X, σ) is a net of σ-continuous operators such that the following assertions are satisﬁed.

(AS1) There exists M > 0 such that kAαk ≤ M for all α ∈ Λ.

σ 0 σ0 0 (AS2) Aαx ∈ co (S x) and Aαy ∈ co (S y) for all α ∈ Λ, x ∈ X and y ∈ Y . (AS3) For every S ∈ S and all x ∈ X and y ∈ Y one has that

lim Aα(S − I)x = lim(S − I)Aαx = 0 α α and

0 0 0 0 lim Aα(S − I)y = lim(S − I)Aαy = 0 α α in the norm topology of X and Y , respectively.

σ Remark 3.1.2. An easy duality argument shows that Aαx ∈ co (S x) for all α ∈ Λ 0 σ0 0 and x ∈ X if and only if Aαy ∈ co (S y) for all α ∈ Λ and y ∈ Y . Thus, in σ (AS2) it actually suffices to assume that Aαx ∈ co (S x) for all α ∈ Λ and x ∈ X. We should point out that our terminology is somewhat different from that in [20]. In the language of Eberlein, the net Aα would be called a system of almost invariant integrals and a semigroup S possessing such a system would be called ergodic. Moreover, we should note that there is no equicontinuity assumption for the averages Aα with respect to σ or with respect to any other consistent topology. Instead, we assume in (AS1) equicontinuity only with respect to the (in general not consistent) norm topology. On the other hand, in (AS3) we assume convergence in the norm topology, which is a stronger assumption than σ-convergence (and also than convergence with respect to a consistent topology on X). Remark 3.1.3. We will frequently make use of the following observation. If (S , A ) is an average scheme on a norming dual pair (X,Y ) and x ∈ fix(S ), then σ co (S x) = {x} and hence, by (AS2), Aαx = x for all α ∈ Λ. We now give some typical examples of average schemes. Throughout, (X,Y ) denotes a norming dual pair. k Example 3.1.4. Let S := {S : k ∈ N0} be a time-discrete semigroup for an operator operator S ∈ L (X, σ) and denote by

n−1 1 X A := Sk (n ∈ N) n n k=0

28 3.1. Average schemes

1 n its Cesàro averages. Assume that limn→∞ n S x = 0 for all x ∈ X and that 1 0n limn→∞ n S y = 0 for all y ∈ Y . Moreover, assume that there exists M > 0 such that kAnk < M for all n ∈ N, i.e. S is Cesàro bounded. Both assumptions are n satisfied if S is power-bounded, i.e. supn∈NkS k < ∞. Clearly, (AS1) and (AS2) are satisfied. As for (AS3), we have

1 n lim An(S − I)x = lim (S − I)x = 0 for all x ∈ X n→∞ n→∞ n 0 0 and, similarly, limn→∞ An(S − I)y = 0 for all y ∈ Y . Thus, (S , (An)n∈N) is an average scheme. k Example 3.1.5. We again consider S := {S : k ∈ N0} for an operator S ∈ 1 n n L (X, σ). If S has spectral radius r(S) = limn→∞kS k ≤ 1, then for r ∈ [0, 1) P∞ k k the series k=0 r S converges in operator norm and thus represents an element of L (X, σ). We denote by

∞ X k k Arx := (1 − r) r S x (r ∈ [0, 1)) k=0 the Abel averages of S. If M := sup0≤r<1kArk < ∞, then S is called Abel bounded. Note that power-bounded operators are Abel bounded. For every Abel bounded operator S ∈ L (X, σ), the pair (S , (Ar)r∈[0,1)) is an average scheme. Indeed, (AS1) is clear. As for (AS2) we see that

n 1 − r X 1 A − rkSk ≤ 1 − M → 0 (n → ∞). r 1 − rn+1 1 − rn+1 k=0

Hence Arx belongs even to the norm closure of co(S x). It remains to verify (AS3). To this end, note that Abel’s summation by parts yields

n n−1 X X lim rk(Sk+1x − Skx) = lim (Sn+1x − x)rn+1 + (rk − rk+1)(Sk+1x − x) n→∞ n→∞ k=0 k=0 n−1 X = lim (1 − r) rkSk+1x − (1 − rn)x n→∞ k=0

= ArSx − x for all x ∈ X and r ∈ [0, 1). Thus,

kArSx − Arxk = (1 − r)kArSx − xk ≤ (1 − r)(MkSk + 1)kxk → 0 as r ↑ 1. On Y , one argues similarly.

29 3. Mean ergodic theorems on norming dual pairs

Example 3.1.6. Let S := (S(t))t≥0 ⊂ L (X, σ) be a time-continuous semigroup that is integrable on (X,Y ), cf. [47]. This means that there exists M ≥ 1 and ω ∈ R such that kS(t)k ≤ Meωt for all t ≥ 0. Moreover, for all x ∈ X and y ∈ Y the function t 7→ hS(t)x, yi is measurable and for some (equivalently, all) λ ∈ C with Re λ > ω there exists an operator R(λ) ∈ L (X, σ) such that Z ∞ hR(λ)x, yi = e−λthS(t)x, yi dy 0 for all x ∈ X and y ∈ Y . It follows from [47, Thm 5.8] that if S is an integrable semigroup, then for every t > 0 there exists an operator At ∈ L (X, σ) such that 1 Z t hAtx, yi = hS(s)x, yi ds. t 0

We call the semigroup S Ces`aro bounded if M := supt>0kAtk < ∞. If S is an 1 1 0 integrable and Ces`arobounded semigroup such that t S(t)x → 0 and t S(t) y → 0 as t → ∞ for arbitrary x ∈ X and y ∈ Y , then (S , (At)t>0) is an average scheme. (AS1) is clear and (AS2) can be deduced from the Hahn-Banach theorem on the 0 σ locally convex spaces (X, σ) and (Y, σ ) as follows. Assume that Atx 6∈ co (S x) for some x ∈ X. Then there exists y ∈ (X, σ)0 = Y and ε > 0 such that

Re hAtx, yi + ε ≤ Re hS(s)x, yi for all s ≥ 0. This implies that 1 Z t Re hAtx, yi + ε = Re hS(s)x, yi ds + ε ≥ Re hAtx, yi + 2ε t 0 and the contradiction ε ≤ 0 follows. In order to verify (AS3), we note that for all t, s > 0 we have s s A − A S(s) = (I − S(t))A = A (I − S(t)) t t t s t s as is easy to see using the semigroup law. Consequently, for every x ∈ X and s > 0 we have −1 −1 kAtx − AtS(s)xk ≤ sM(kxkt + kt S(t)xk) → 0 as t → ∞. On Y , one argues similarly. Finally, we remark that (S , (At)t>0) is in particular an average scheme whenever the integrable semigroup S is bounded. Concerning the last example, let us remark that if S is an integrable semigroup, then the operators R(λ) form a pseudo resolvent. Hence, there is a unique possibly multivalued operator G with R(λ) = (λ − G )−1, the generator of S . In this case, as a consequence of [47, Prop 5.7], ﬁx(S ) = ker G = {x ∈ X :(x, 0) ∈ G }. For more information about integrable semigroups and their generators, we refer to [47].

30 3.2. An ergodic theorem on norming dual pairs

3.2 An ergodic theorem on norming dual pairs

In the following we prove a mean ergodic theorem for average schemes on norming dual pairs. We start with the deﬁnition of weak ergodicity.

Deﬁnition 3.2.1. We say that an average scheme (S , A ) on a norming dual pair (X,Y ) is weakly ergodic if the σ-limit of (Aαx) exists for every x ∈ X and 0 0 the σ -limit of (Aαy) exists for every y ∈ Y . In the mean ergodic theorem on norming dual pairs we need a slightly stronger version of assertion (ii) of Theorem 3.0.9. This is due to the fact that the strategy for the proof diﬀers from the classical one since not all assertions corresponding to (i) – (iv) are equivalent in our situation. We use the following terminology.

Deﬁnition 3.2.2. We say that a net (xα)α∈Λ in a topological space Ω clusters if every subnet of (xα) has a cluster point, i.e. it has a convergent subnet. Remark 3.2.3. A net clusters whenever the set of its elements is relatively compact. However, if a net (xα)α∈Λ clusters, one cannot infer that the set {xα : α ≥ α0} is relatively compact for some α0 ∈ Λ. For a sequence, these two properties are equivalent, which is a consequence of [40, Lem 5.4]. The following is the main result of this section.

Theorem 3.2.4. Let (S , A ) be an average scheme on a norming dual pair (X,Y ). Then the following assertions are equivalent.

(i) The average scheme (S , A ) is weakly ergodic.

(ii) For every x ∈ X the net (Aαx) clusters in (X, σ) and for every y ∈ Y the 0 0 net (Aαy) clusters in (Y, σ ). If these equivalent conditions are satisﬁed, then

(iii) The fixed spaces fix(S ) and fix(S 0) separate each other.

0 (iv) We have X = ﬁx(S )⊕spanσ rg(I −S ) and Y = ﬁx(S 0)⊕spanσ rg(I −S 0).

(v) The operator P , defined by P x := σ- limα Aαx, belongs to L (X, σ) and the 0 0 0 0 σ-adjoint P of P is given by P y = σ - limα Aαy for all y ∈ Y . Moreover, P is the projection onto fix(S ) along spanσ rg(I − S ), P 0 the projection 0 onto fix(S 0) along spanσ rg(I − S 0) and PS = SP = P for all S ∈ S . For a weakly ergodic average scheme (S , A ), the operator P from (v) is called the ergodic projection. Note that the ergodic projection P is uniquely determined by the semigroup S and independent of the averages A . We prepare the proof of Theorem 3.2.4 through a series of lemmas.

31 3. Mean ergodic theorems on norming dual pairs

Lemma 3.2.5. Let X be a Banach space and S ⊂ L (X) be semigroup of bounded operators on X. Moreover, let (Aα)α∈Λ ⊂ L (X) be a net and let x ∈ X be such that lim(S − I)Aαx = 0 for all S ∈ S . α Assume that Z ⊂ X∗ separates the points in X and S∗Z ⊂ Z for all S ∈ S . Then every σ(X,Z)-cluster point of (Aαx) belongs to ﬁx(S ).

Proof. Let w be a σ(X,Z)-cluster point of (Aαx) and ﬁx S ∈ S . We have

Sw − w = (S − I)(w − Aαx) + (S − I)Aαx

for all α ∈ Λ and, by assumption, (S − I)Aαx → 0 in norm and hence also with respect to σ(X,Z). Now fix z ∈ Z. Given ε > 0, we find α0 such that |h(S − I)Aαx, zi| < ε for all ∗ α ≥ α0. Since S z ∈ Z and since w is a σ(X,Z)-cluster point of (Aαx), we find some β ≥ α0 such that

∗ |h(S − I)(w − Aβx), zi| = |hw − Aβx, S z − zi| ≤ ε.

This implies that |hSw − w, zi| ≤ 2ε. Since ε > 0 was arbitrary, it follows that hSw − w, zi = 0 and thus, since z ∈ Z was arbitrary, w = Sw.

Lemma 3.2.6. Let (S , A ) be an average scheme on a norming dual pair (X,Y ). Then X0 := {x ∈ X : lim Aαx exists w.r.t. k · k} α is a norm-closed subspace of X and invariant under the action of S . Moreover, the k · k k · k sum fix(S ) + span rg(I − S ) is direct and fix(S ) ⊕ span rg(I − S ) ⊂ X0. Finally, P0x := k · k- limα Aαx defines a bounded operator on X0 which is a k · k projection onto fix(S ) with span rg(I − S ) ⊂ ker P0.

Proof. Let xk ∈ X0 and lim xk = x with respect to k · k. Then P0xk ∈ ﬁx(S ) by Lemma 3.2.5 for all k ∈ N and we have

kP0xk − P0xlk = limkAα(xk − xl)k ≤ Mkxk − xlk α

for all k, l ∈ N where M is such that kAαk ≤ M for all α ∈ Λ. Since (xk) is a Cauchy sequence, so is (P0xk). Thus, P0xk → x¯ for some x¯ which belongs to fix(S ) as the latter is closed. A 3ε-argument shows that Aαx → x¯. It follows that X0 is closed and P0x =x ¯. We have seen that kP0k ≤ M and P0X0 ⊂ fix(S ). Conversely, fix(S ) ⊂ P0X0 since Aαx ≡ x = P0x for x ∈ fix(S ). Hence, P0X0 = fix(S ) and P0 is a projection. The S -invariance of X0 follows from (AS3).

32 3.2. An ergodic theorem on norming dual pairs

By the definition of an average scheme, limα Aαx = 0 for all x ∈ rg(I − S ). In view of the uniform boundedness of the operators Aα, this remains true for k · k x ∈ span rg(I − S ). Since Aαx → x for x ∈ fix(S ), it follows that the sum of fix(S ) and spank · k rg(I − S ) is direct and that fix(S ) ⊕ spank · k rg(I − S ) ⊂ X0.

Lemma 3.2.7. Let (S , A ) be an average scheme on a norming dual pair (X,Y ). If fix(S ) separates fix(S 0), then fix(S ) + rg(I − S ) is σ(X,Y )-dense in X. If fix(S 0) separates fix(S ), then the sum fix(S ) + spanσ rg(I − S ) is direct.

Proof. Assume that fix(S ) separates fix(S 0). Let y ∈ Y be such that hx, yi = 0 for all x ∈ fix(S ) + rg(I − S ). Then, in particular, 0 = hx − Sx, yi = hx, y − S0yi for all x ∈ X and S ∈ S . Since X separates Y , it follows that y = S0y for all S ∈ S , i.e. y ∈ fix(S 0). Moreover, hx, yi = 0 for all x ∈ fix(S ). By assumption, this implies y = 0. It now follows from the Hahn-Banach theorem, applied on the locally convex space (X, σ), that fix(S ) + rg(I − S ) is σ(X,Y )-dense in X. Now assume that fix(S 0) separates fix(S ). Since every y ∈ fix(S 0) vanishes on rg(I − S ), it also vanishes on spanσ rg(I − S ) by linearity and continuity. Thus, hx, yi = 0 for all x ∈ fix(S ) ∩ spanσ rg(I − S ) and y ∈ fix(S 0). As fix(S 0) separates fix(S ), it follows that 0 is the only element of fix(S ) ∩ spanσ rg(I − S ).

Lemma 3.2.8. Let (S , A ) be an average scheme on a norming dual pair (X,Y ) and τ be a locally convex topology on X ﬁner than σ. Let

X1 := {x ∈ X :(Aαx)α∈Λ clusters in (X, τ)}

0 and assume that ﬁx(S ) separates ﬁx(S ). Then τ- limα Aαx ∈ X exists for all x ∈ X1.

Proof. Applying Lemma 3.2.6 to the average scheme (S 0, A 0) on (Y,X), we find that 0 Y0 := {y ∈ Y : k · k- lim Aαy exists} α is a norm-closed subspace of Y that contains fix(S 0) ⊕ rg(I − S 0). Moreover, 0 there exists an operator R0 ∈ L (Y0) such that k · k- limα Aαy = R0y for all y ∈ Y0. 0 As fix(S ) separates fix(S ), Lemma 3.2.7 implies that Y0 is σ(Y,X)-dense in ∗ Y . Hence, we may identify X with a subspace of Y0 . Then it follows that ∗ ∗ σ(Y0 ,Y0)- limα Aαx = R0x for all x ∈ X. Let us fix x ∈ X1 and choose an arbitrary subnet of (uβ) of (Aαx). By assumption, (uβ) has a τ-cluster point ∗ ∗ x¯ ∈ X. Since x¯ is also a σ(Y0 ,Y0)-cluster point of (Aαx), we infer that x¯ = R0x. ∗ Thus, every subnet of (Aαx) has a subnet converging to R0x ∈ X in (X, τ). This ∗ implies that τ- limα Aαx = R0x for all x ∈ X1.

33 3. Mean ergodic theorems on norming dual pairs

Now, we have the tools at hand to prove Theorem 3.2.4.

Proof of Theorem 3.2.4. The implication (i) ⇒ (ii) is trivial, so assume that (ii) holds. Let us verify (iii) first. Since Y is norming for X, given x ∈ fix(S ), x 6= 0, 0 we find y ∈ Y such that hx, yi = a 6= 0. By assumption, Aαy has a σ(Y,X)-cluster point z which, by Lemma 3.2.5, is an element of fix(S 0). Since

0 hx, Aαyi = hAαx, yi = hx, yi = a

for all α ∈ Λ, it follows that hx, zi = a =6 0. Hence, fix(S 0) separates fix(S ). Interchanging the roles of X and Y , it follows that fix(S ) separates fix(S 0). Now, Assertion (i) follows immediately from Lemma 3.2.8 applied to the average schemes (S , A ) and (S 0, A 0) and the weak topologies σ(X,Y ) and σ(Y,X), respectively. We continue with the verification of Assertion (iv). By (iii) and Lemma 3.2.7, the sums

0 ﬁx(S ) + spanσ rg(I − S ) and ﬁx(S 0) + spanσ rg(I − S 0)

are direct and dense in X and Y with respect to σ and σ0, respectively. Let x ∈ X and x¯ := limα Aαx ∈ ﬁx(S ). Since x − co(S x) ⊂ span rg(I − S ), we have that σ x − Aαx ∈ span rg(I − S ) for all α ∈ Λ and, consequently,

σ x − x¯ = σ- lim(x − Aαx) ∈ span rg(I − S ). (3.2.1) α

This shows that X = fix(S ) ⊕ spanσ rg(I − S ) and analogously we deduce that 0 Y = fix(S 0) ⊕ spanσ rg(I − S 0). In order to verify Assertion (v), we consider the operators P and R defined by 0 P x := σ- limα Aαx and Ry := σ- limα Aαy. Since sup{kAαk : α ∈ Λ} < ∞ and X and Y are norming for each other, it follows that P ∈ L (X) and R ∈ L (Y ). Moreover, for x ∈ X and y ∈ Y we have

0 hP x, yi = limhAαx, yi = limhx, Aαyi = hx, Ryi. α α

This implies that P ∗Y = RY ⊂ Y , hence P ∈ L (X, σ). As fixed points are invariant under A , it follows from Lemma 3.2.5 that P is a projection with rg P = fix(S ) and, by (AS3), span rg(I − S ) ⊂ ker P . Since P is σ-continuous and ker P is closed, spanσ rg(I − S ) ⊂ ker P . The converse inclusion follows from (3.2.1). Interchanging the roles of X and Y , we see that P 0 = R is the projection 0 onto fix(S 0) along spanσ (I − S 0). In view of SP x = P x and P (S − I)x = limα Aα(S − I)x = 0 for all x ∈ X and S ∈ S , P commutes with every operator in S .

34 3.2. An ergodic theorem on norming dual pairs

Theorem 3.2.4 is symmetric in X and Y , in that for every statement concerning X and A there is a corresponding statement about Y and A 0. This symmetry ∗ is crucial. Indeed, in the case where Y = X , the norm-bounded net (Aαx) is always relatively σ-compact and hence σ-clusters. However, it does not necessarily σ-converge as the example of the left shift on `∞ shows. 0 Moreover, even if Aαx σ-converges for all x ∈ X, one cannot deduce σ -conver- 0 gence of Aαy for all y ∈ Y and hence no weak ergodicity of the average scheme, see Example 3.4.1. Comparing Theorem 3.2.4 with the classical mean ergodic theorem on Banach spaces, an immediate question is whether assertions (i) and (ii) are also equivalent with (iii) and (iv). This is not the case in general as Examples 3.4.2 and 3.4.3 show. However, some weaker results hold true, which are stated in the following proposition.

Proposition 3.2.9. Let (S , A ) be an average scheme on a norming dual pair (X,Y ).

(a) Suppose that fix(S ) and fix(S 0) separate each other. If fix(S ) (hence also fix(S 0)) is finite dimensional, then

0 X = ﬁx(S )⊕spanσ rg(I −S ) and Y = ﬁx(S 0)⊕spanσ rg(I −S 0). (3.2.2)

(b) Now assume that (3.2.2) holds and let P denote the projection onto ﬁx(S ) along spanσ rg(I − S ). Then P ∈ L (X, σ), P 0 is the projection onto ﬁx(S 0) 0 along spanσ rg(I − S 0) and

0 0 σ(X,Y0)- lim Aαx = P x and σ(Y,X0)- lim Aαy = P y α α

k · k for all x ∈ X and y ∈ Y , where X0 := fix(S ) ⊕ span rg(I − S ) and 0 k · k 0 0 Y0 := fix(S ) ⊕ span rg(I − S ). Moreover, fix(S ) and fix(S ) separate each other.

Proof. (a) By Lemma 3.2.7, the sum fix(S ) ⊕ spanσ rg(I − S ) is direct and σ-dense in X. Since fix(S ) is finite dimensional, the sum is σ-closed by [43, §15.5(3)]. It follows that the sum equals X. Similarly one sees that Y = fix(S 0) ⊕ spanσ rg(I − S 0). (b) For x ∈ X we have x = P x+(I−P )x ∈ fix(S )⊕spanσ rg(I−S ). To prove Aαx → P x with respect to σ(X,Y0), it suffices to show that limαhAαx, yi = 0 for σ x ∈ span rg(I − S ) and y ∈ Y0 as Aαx ≡ x for every x ∈ fix(S ). To that end, first note that for w = (I − S)v ∈ rg(I − S ) and y ∈ fix(S 0) we have that

hw, yi = h(I − S)v, yi = hv, (I − S0)yi = 0,

35 3. Mean ergodic theorems on norming dual pairs

i.e. y vanishes on rg(I − S ) and hence on spanσ rg(I − S ) by continuity. Now let y = z − S0z ∈ rg(I − S 0). Then

limhAαx, yi = limh(I − S)Aαx, zi = 0 α α

by (AS3). In view of the uniform boundedness of (Aα), property (AS1), it follows k · k 0 that limαhAαx, yi = 0 for y ∈ span rg(I − S ), too. Altogether, we have proven that σ(X,Y0)- lim Aαx = P x. α It is easy to see that P is norm-closed, hence it is bounded by the closed graph theorem. 0 By interchanging the roles of X and Y , one sees that σ(Y,X0)- limα Aαy = Ry 0 where R ∈ L (Y ) is the projection onto fix(S 0) along spanσ rg(I − S 0). 2 Now let x ∈ X and y ∈ Y be given. Since P x = P x ∈ fix(S ) ⊂ X0 and 2 0 Ry = R y ∈ fix(S ) ⊂ Y0, it follows that

2 0 hP x, yi = hP x, yi = limhAαP x, yi = limhP x, Aαyi α α 0 = hP x, Ryi = limhAαx, Ryi = limhx, AαRyi = hx, Ryi. α α

∗ 0 ∗ This shows that P Y ⊂ Y and P = P |Y = R. In particular, P ∈ L (X, σ). In order to prove the ﬁnal assertion, let x ∈ ﬁx(S ), x 6= 0, and y ∈ Y such that hx, yi= 6 0. Then

0 6= hx, yi = hP x, yi = hx, P 0yi

shows that fix(S 0) separates fix(S ). Interchanging the roles of X and Y finishes the proof.

Example 3.4.3 shows that in Part (a) of Proposition 3.2.9 the assumption that the ﬁxed spaces are ﬁnite dimensional cannot be omitted.

3.3 An ergodic theorem on the space of measures

Throughout this section, we ﬁx a Polish space Ω and work on the norming dual pair (M (Ω),Cb(Ω)), which seems to be the most interesting norming dual pair for applications. Recall from Section 2.1 that we write σc for the σ(M (Ω),Cb(Ω))- 0 topology on M (Ω) and σc for the σ(Cb(Ω), M (Ω))-topology on Cb(Ω). In the following we break the symmetry between X = M (Ω) and Y = Cb(Ω) by imposing additional assumptions on the (adjoint) semigroup on the function space such that assertions (i) – (iv) of Theorem 3.2.4 are equivalent. In view of Proposition 3.2.9(b), (iv) implies (iii), so the question is whether (iii) implies (i).

36 3.3. An ergodic theorem on the space of measures

Examples 3.4.2 and 3.4.3 show that this is not true without additional assumptions.

Let d be a complete metric generating the topology of Ω and denote by Lipb(Ω, d) the space of all bounded Lipschitz continuous functions on Ω with respect to d. We assume the average scheme (S , A ) to satisfy the following. 0 Hypothesis 3.3.1. For all f ∈ Lipb(Ω, d) the net (Aαf)α∈Λ clusters in (Cb(Ω), β0). A priori, this requirement depends on the choice of the metric d. However, Hypothesis 3.3.1 is necessary for the assertion of Theorem 3.3.9 which, in turn, is independent of the metric d. Hence, under the other assumptions of Theorem 3.3.9, Hypothesis 3.3.1 holds for some metric d if and only if it holds for every complete metric on Ω that generates its topology. Let us fix such a metric d for the rest of this section. Let us compare Hypothesis 3.3.1 with Assertion (ii) of Theorem 3.2.4. Instead 0 0 of assuming that (Aαf) clusters in (Cb(Ω), σc) for every f ∈ Cb(Ω), we require 0 that the net (Aαf) clusters with respect to the finer topology β0, but only for those functions f with some additional regularity, namely for Lipschitz functions. At first sight, Hypothesis 3.3.1 seems to be rather technical. However, as already mentioned, it is necessary for Theorem 3.3.9 and, important from the point of view of applications, it is implied by both the strong Feller property and the e-property, which are well-known assumptions in the study of ergodic properties of Markov chains and semigroups cf. [42, 59]. Let us discuss these relationships, starting with the e-property, before continuing with our general theory.

Deﬁnition 3.3.2. A family T ⊂ L (Cb(Ω)) is said to have the e-property if the orbits {T f : T ∈ T } are equicontinuous for all f ∈ Lipb(Ω, d), i.e. for all x ∈ Ω and ε > 0 there exists a δ > 0 such that |T f(x) − T f(y)| ≤ ε for all T ∈ T whenever d(x, y) < δ. 0 A family T ⊂ L (M (Ω), σc) is said to have the e-property, if {T : T ∈ T } has the e-property. A net (Ti)i∈I ⊂ L (M (Ω), σc) is said the have the eventual e-property if there exists a j ∈ I such that {Ti : i ≥ j} has the e-property. An average scheme (S , A ) on (M (Ω),Cb(Ω)) is said to have the (eventual) e-property if (Aα)α∈Λ has the (eventual) e-property.

As an instructive example, consider the shift semigroup S = (S(t))t≥0 on 0 (M (R),Cb(R)), given by (S(t) f)(x) = f(t + x). Then S has the e-property (we will see below that this implies that also every average scheme (S , A ) has the R e-property), since for f ∈ Lipb( , | · |) we have

|(S(t)0f)(x) − (S(t)0f)(y)| = |f(t + x) − f(t + y)| ≤ L|x − y|

37 3. Mean ergodic theorems on norming dual pairs

for all x, y ∈ R, where L is the Lipschitz constant of f. Note, however, that this 0 does not imply that the orbit S(t) f is equicontinuous for all f ∈ Cb(R). Before giving further examples, let us prove the mentioned result concerning the e-property of S and that of A .

Lemma 3.3.3. Let S ⊂ L (M (Ω), σc) be a semigroup which has the e-property. Then every average scheme (S , A ) has the e-property.

Proof. Let f ∈ Lipb(Ω, d), x ∈ Ω and ε > 0. By assumption, there exists δ > 0 such that |(S0f)(x) − (S0f)(y)| < ε for all S ∈ S whenever d(x, y) < δ. Pick such a y ∈ Ω. Pn 0 0 By (AS2), given α ∈ Λ we ﬁnd a function gy := k=1 akSkf ∈ co(S f) such 0 that |hAαf − gy, δx − δyi| ≤ ε. It follows that

0 0 0 |(Aαf)(x) − (Aαf)(y)| ≤ |hAαf − gy, δx − δyi| + |hgy, δx − δyi| n X 0 0 ≤ ε + ak|Skf(x) − Skf(y)| ≤ 2ε. k=1

0 Since ε and α were arbitrary, {Aαf : α ∈ Λ} is equicontinuous. Thus, (S , A ) has the e-property.

Example 3.4.2 shows that if S does not have the e-property, there might be A and Afsuch that (S , A ) has the e-property whereas (S , Af) does not have the e-property.

Remark 3.3.4. In view of the Arzel`a-Ascolitheorem [41, Thm 3.6] and (AS1), an average scheme (S , A ) has the (eventual) e-property if and only if for every 0 0 f ∈ Lipb(Ω, d) the set {Aαf : α ∈ Λ} (the set {Aαf : α ≥ α0}) is relatively β0- compact, equivalently, the set is relatively compact in the compact-open topology. Thus, an average scheme with eventual e-property satisﬁes Hypothesis 3.3.1. We next discuss the strong Feller property which also implies Hypothesis 3.3.1.

Deﬁnition 3.3.5. Let T ∈ L (M (Ω), σb) be a weakly continuous operator. If its 0 adjoint satisﬁes T f ∈ Cb(Ω) for all f ∈ Bb(Ω), then T is called strong Feller. If the family 0 {T f : f ∈ Bb(Ω), |f| ≤ c1} is even equicontinuous for all c > 0, then T is said to be ultra Feller.

Let us recall from Section 2.1 that an operator T ∈ L (M (Ω), σc) is in particular weakly continuous in the sense of Deﬁnition 2.2.4 and hence is given by a transition kernel. Thus, its adjoint has a unique extension to an operator on Bb(Ω).

38 3.3. An ergodic theorem on the space of measures

Proposition 3.3.6. Let S ⊂ L (M (Ω), σc) be a semigroup such that some operator S ∈ S is strong Feller. Then every average scheme (S , A ) satisﬁes Hypothesis 3.3.1.

Proof. Let f ∈ Cb(Ω) and (S , A ) be an average scheme. Since S is strong Feller, the operator S2 ∈ S is ultra Feller [52, §1.5]. Hence, by (AS1) and the Arzel`a- 02 0 Ascoli theorem [41, Thm 3.6], the set {S Aαf : α ∈ Λ} is relatively β0-compact. 0 0 02 0 Thus, every subnet of (Aαf) has a subnet (Aα(β)f)β∈J such that (S Aα(β)f)β∈J converges in (Cb(Ω), β0). By (AS3),

02 0 0 S Aα(β)f − Aα(β)f → 0 in norm and thus with respect to the strict topology which is coarser.

Now we return to our main line of study. Besides Hypothesis 3.3.1, we will impose further assumptions. First of all, we assume that S is a Markovian semigroup, i.e. every operator S ∈ S is Markovian by which we mean that S is ∗ positive and S 1 = 1. If kS denotes the transition kernel associated with S, this is equivalent with the requirement that kS(x, · ) is a probability measure for all x ∈ Ω and S ∈ S . Since the applications we have in mind for our theory concern transition semigroups of Markov chains or Markov processes, this assumption is rather natural. We will call an average scheme (S , A ) Markovian if S is Markovian. It follows from (AS2) that this implies that every operator Aα is Markovian, too. Second, we assume that the directed index set Λ of the averages (Aα)α∈Λ in this section has a coﬁnal subsequence. This holds, for instance, in the classical situations of Examples 3.1.4, 3.1.5 and 3.1.6.

We start with two auxiliary lemmas on β0-equicontinuous operators. Let us recall that a family T of linear operators on a locally convex space (X, τ) is called τ-equicontinuous, if for every τ-continuous seminorm p, there exists a τ-continuous seminorm q such that p(T x) ≤ q(x) for every T ∈ T and x ∈ X. In the case where (X, τ) = (Cb(Ω), β0), we have the following characterization of β0-equicontinuous operators, see [46, Prop 4.2]. A family T ⊂ L (Cb(Ω), β0) is β0-equicontinuous if and only if for every compact set K ⊂ Ω and every ε > 0 there exists a compact set L ⊂ Ω such that |kT |(x, Ω\L) ≤ ε for all x ∈ K and T ∈ T , where kT denotes the transition kernel of T . In what follows we use that a family of Borel measures on Ω is relatively σ- compact if and only if it is uniformly tight and uniformly bounded in the variation norm, cf. Theorems 8.6.7. and 8.6.8. of [15]. Recall that a family F ⊂ M (Ω) of Borel measures is said to be uniformly tight if for all ε > 0 there exists a compact set K ⊂ Ω such that |µ|(Ω\K) < ε for all µ ∈ F .

39 3. Mean ergodic theorems on norming dual pairs

0 Lemma 3.3.7. Let {Tj : j ∈ J} ⊂ L (Cb(Ω), σc) be a family of Markovian operators that has the e-property. Suppose that the family {kj(x, · ): j ∈ J} is uniformly tight for all x ∈ Ω, where kj denotes the transition kernel associated with Tj. Then the operators {Tj : j ∈ J} are β0-equicontinuous.

Proof. Let K ⊂ Ω be compact and ε > 0. By assumption, for every x ∈ Ω there exists a compact set Lx ⊂ Ω such that kj(x, Ω\Lx) ≤ ε for all j ∈ J. We denote by ε Lx := {y ∈ Ω : dist(y, Lx) < ε}

the open ε-neighborhood of Lx and deﬁne

dist(y, Ω\Lε ) : x fx(y) = ε . dist(y, Lx) + dist(y, Ω\Lx)

Then f is Lipschitz continuous and 1 ≤ f ≤ 1 ε . Since the family (T ) has x Lx x Lx j the e-property, there exists δx > 0 such that

|(Tjfx)(x) − (Tjfx)(y)| < ε

for all j ∈ J whenever d(x, y) < δx. By compactness of K, we ﬁnd x1, . . . , xn ∈ Ω such that n [ K ⊂ {y ∈ Ω: d(xk, y) < δxk }. k=1 Let L := ∪n L and f(y) := max{f (y), . . . , f (y)}. Since Lε = ∪n Lε we k=1 xk x1 xn k=1 xk have 1L ≤ f ≤ 1Lε . Now ﬁx an arbitrary x ∈ K and choose k ∈ {1, . . . , n} such

that d(x, xk) < δxk . Then we have Z ε kj(x, L ) ≥ f(y)kj(x, dy) = (Tjf)(x) ≥ (Tjfxk )(xk) − ε Ω Z

= fxk (y)kj(xk, dy) − ε ≥ kj(xk,Lxk ) − ε ≥ 1 − 2ε Ω for every j ∈ J. It follows from [24, Thm 3.2.2] that the family

{kj(x, · ): j ∈ J, x ∈ K}

is uniformly tight. Since K ⊂ Ω was arbitrary, the operators Tj are β0-equicontinuous by [46, Prop 4.2].

Lemma 3.3.8. Let {Tj : j ∈ J} ⊂ L (Cb(Ω), β0) be a β0-equicontinuous family of operators with e-property. Then {Tjf : j ∈ J} is equicontinuous for all f ∈ Cb(Ω).

40 3.3. An ergodic theorem on the space of measures

Proof. Since Lipb(Ω, d) is a subalgebra of Cb(Ω) which separates the points of Ω, it follows from the Stone-Weierstrass theorem that Lipb(Ω, d) is dense in (Cb(Ω), β0), see [25, Theorem 11]. Fix f ∈ Cb(Ω) and xn, x ∈ Ω with lim xn = x. We show that (Tjf)(xn) converges to (Tjf)(x), uniformly in j ∈ J. This proves the equicontinuity of {Tjf : j ∈ J} in x. Consider the compact set K := {x}∪{xn : n ∈ N} and the associated seminorm 1 p(h) := sup{|h(x)| : x ∈ K} = khϕk∞ for ϕ = K . Since the family (Tj) is β0- equicontinuous, there exists a β0-continuous seminorm q : Cb(Ω) → [0, ∞) such that p(Tjh) ≤ q(h) for all h ∈ Cb(Ω) and j ∈ J. Now given ε > 0, pick g ∈ Lipb(Ω, d) such that q(f − g) ≤ ε. Since the family {Tjg : j ∈ J} is equicontinuous, there exists n0 ∈ N such that

|(Tjg)(xn) − (Tjg)(x)| ≤ ε for all n ≥ n0 and all j ∈ J. This implies that

|(Tjf)(xn) − (Tjf)(x)| ≤ 2q(f − g) + |(Tjg)(xn) − (Tjg)(x)| ≤ 2ε + ε for all n ≥ n0 and j ∈ J. Now we prove the main result of this section.

Theorem 3.3.9. Let (S , A ) be a Markovian average scheme on the norming dual pair (M (Ω),Cb(Ω)) that satisﬁes Hypothesis 3.3.1 and suppose that there exists an increasing coﬁnal sequence in Λ. Then the following assertions are equivalent.

0 0 (i) (S , A ) is weakly ergodic and β0- limα Aαf = P f for all f ∈ Cb(Ω) where P is the ergodic projection.

(ii) For every x ∈ Ω the net (Aαδx) has a σc-cluster point.

(iii) ﬁx(S ) separates ﬁx(S 0).

(iv) M (Ω) = fix(S ) ⊕ spanσ(I − S ). Proof. By Theorem 3.2.4, (i) implies (ii) – (iv). Moreover, the implication (iv) ⇒ (iii) is part of the proof of Proposition 3.2.9 and (iii) follows from (ii) as in the proof of Theorem 3.2.4. Hence, it remains to prove that (iii) implies (i) to complete the proof. 0 Let us assume that fix(S ) separates fix(S ). We denote by (αn) ⊂ Λ an 0 increasing cofinal sequence. As (Aαf) clusters in (Cb(Ω), β0), Lemma 3.2.8 implies that 0 0 β0- lim Aαf = β0- lim Aα f ∈ Cb(Ω) α n→∞ n

41 3. Mean ergodic theorems on norming dual pairs

exists for all f ∈ Lipb(Ω, d). Fix a non-negative measure µ ∈ M (Ω). Then the scalar sequence hf, A µi = hA0 f, µi converges as n → ∞ for all f ∈ Lip (Ω, d). αn αn b N Thus, by [15, Cor 8.6.3], the family {Aαn µ : n ∈ } is uniformly tight and Prohorov’s theorem [15, Thm 8.6.2] implies that, passing to a subsequence,

(Aαn µ)n∈N converges weakly to some measure µ˜ ∈ M(Ω). Altogether, this shows that

0 0 limhf, Aαµi = limhAαf, µi = lim hAα f, µi = lim hf, Aαn µi = hf, µ˜i α α n→∞ n n→∞

for all f ∈ Lipb(Ω, d). Since Ω is separable, the set Lipb(Ω, d) is convergence deter- mining by [24, Prop 3.4.4], hence it follows that σ- limα Aαµ = µ˜. Decomposing a general measure in positive and negative part shows that σ- limα Aαµ ∈ M (Ω) exists for every µ ∈ M (Ω). By Hypothesis 3.3.1 and Remark 3.2.3, the set {A0 f : n ∈ N} is relatively αn β0-compact for every f ∈ Lipb(Ω, d), which implies by the Arzel`a-Ascolitheorem N (cf. Remark 3.3.4) that the family {Aαn : n ∈ } has the e-property. Now, we infer N from Lemma 3.3.7 that the operators {Aαn : n ∈ } are β0-equicontinuous. By Lemma 3.3.8 this implies that the orbits {A0 f : n ∈ N} are equicontinuous for all αn f ∈ C (Ω). Using the Arzel`a-Ascolitheorem again, it follows that {A0 f : n ∈ N} b αn is relatively β0-compact for all f ∈ Cb(Ω).

Now we conclude from Theorem 3.2.4 that the average scheme (S , (Aαn )) is weakly ergodic with an ergodic projection P ∈ L (M (Ω), σc). Since (αn) was

arbitrary and P does not depend on the averages (Aαn ), even (S , A ) is weakly 0 ergodic. Finally, the β0-convergence of (Aαf) follows from Lemma 3.2.8.

Remark 3.3.10. Assume that Λ = N or Λ = [0, ∞) in their natural ordering and 0 N that α 7→ Aαf is continuous as a map with values in (Cb(Ω), β0). If Λ = , this is always the case, for Λ = [0, ∞) this is true for the Cesàroaverages At of 0 a semigroup S = (S(t))t≥0 on (M (Ω),Cb(Ω)) such that S has β0-continuous 0 0 orbits. If (S , A ) is weakly ergodic, and Aαf converges to P f with respect to 0 β0, then the means {Aα : α ∈ Λ} are β0-equicontinuous. Indeed, in this case the function F : Λ ∪ {∞} → L (Cb(Ω), σ), defined by F (α) = Aα for α 6= ∞ and F (∞) = P is strongly β0-continuous. Hence, the equicontinuity follows from [46, Lemma 3.8]. We are thus in the situation of mean ergodic theorems on locally convex spaces [20, 53]. However, note that we do not a priory assume β0-equicontinuity in Theorem 3.3.9 since, as the example of the shift semigroup shows, Hypothesis 3.3.1 alone does not imply equicontinuity of the averages. Remark 3.3.11. It is immediate that, in the situation of Theorem 3.3.9, if (S , A ) is weakly ergodic, then any average scheme (S , B) which satisfies Hypothesis 3.3.1 is also weakly ergodic.

42 3.4. Counterexamples

3.4 Counterexamples

We conclude this chapter with a collection of examples that illustrate the optimality of the results obtained in Section 3.2. Our first example shows that even if Aαx 0 σ-converges for all x ∈ X, it can happen that on Y the averages Aαy do not σ0-converge for some y ∈ Y . 1 Example 3.4.1. Consider the norming dual pair (` , c0) and the power-bounded 1 1 operator S : ` → ` , defined by S : (t1, t2, t3,...) 7→ (t1 + t2, t3,...). Then ∗ the adjoint operator is given by S (s1, s2,...) = (s1, s1, s2,...). In particular, ∗ 1 S c0 ⊂ c0 so that S ∈ L (` , σ). Since n n X S (t1, t2,...) = tj, tn+1, tn+2,... j=1 P∞ clearly σ-converges to ( j=1 tj, 0, 0,...), the σ-limit of the Cesàroaverages

n−1 1 X A x = Skx n n k=0 exists for all x ∈ `1. However, in this situation the ergodic projection

∞ X P :(t1, t2, t3,... ) 7→ tj, 0, 0,... j=1

1 1 does not respect the duality, i.e. P 6∈ L (` , σ). Indeed, for x = (tj) ∈ ` and ∞ 1 ∗ y = (sj) ∈ ` = (` ) we have

∞ X hP x, yi = s1 tj = hx, s11i j=1

∗ 0 0 whence P c0 6⊂ c0. By Theorem 3.2.4, the σ -limit of the adjoint averages Any does not exist for some y ∈ c0. Our next example shows that if (S , A ) is an average scheme so that both X and Y can be decomposed as in (iv) of Theorem 3.2.4, the average scheme is not necessarily weakly ergodic. In fact, we present two diﬀerent averages A and A˜ for the same semigroup S such that (S , A ) is weakly ergodic whereas for (S , A˜) only the weaker convergence properties of Proposition 3.2.9 (b) hold. Example 3.4.2. We consider the set Ω = Z ∪ {∞}, where every point in Z is isolated, whereas the neighborhoods of the extra point ∞ are exactly the sets which contain a set of the form {n, n + 1,...} ∪ {∞} for some n ∈ Z. Note that

43 3. Mean ergodic theorems on norming dual pairs

N 1 N Ω is homeomorphic with {−n : n ∈ } ∪ {1 − n : n ∈ } ∪ {1} endowed with the topology inherited from R, thus Ω is Polish. We work on the norming dual pair (M (Ω),Cb(Ω)). Note that a function f : Ω → R is continuous if and only if limn→∞ f(n) = f(∞) and that M (Ω) = `1(Ω). Consider the semigroup S := {Sn : n ∈ Z} where ( f(k + 1) k ∈ Z (S0f)(k) := f(∞) k = ∞.

0 Then fix(S ) = {c1Ω : c ∈ R} and fix(S ) = {cδ∞ : c ∈ R}. In particular, the fixed spaces separate each other and are finite dimensional so that, as a consequence of Proposition 3.2.9 (a), we have

0 σc 0 σ 0 M (Ω) = ﬁx(S ) ⊕ span rg(I − S ) and Cb(Ω) = ﬁx(S ) ⊕ span c rg(I − S ).

We now consider the average schemes An and A˜n, deﬁned by

n−1 n−1 1 X 1 X A x := Skx and A˜ x := S−kx. n n n n k=0 k=0

That An and Añ are indeed average schemes is proved as in Example 3.1.4. k · k Defining Y0 := fix(S ) ⊕ span rg(I − S ), it follows from Proposition 3.2.9 (b) that 0 ˜0 0 1 σ(Cb(Ω),Y0)- lim Anf = σ(Cb(Ω),Y0)- lim Anf = P f = f(∞) Ω n→∞ n→∞

for all f ∈ Cb(Ω). Actually, using that f(n) → f(∞) as n → ∞ for all f ∈ Cb(Ω), 0 1 it is easy to see that Anf → f(∞) Ω pointwise for all f ∈ Cb(Ω), hence with 0 ˜0 0 1 respect to σc. However, Anf does not σc-converge to f(∞) Ω for some f ∈ Cb(Ω). 1 ˜0 Indeed, for f := N∪{∞} ∈ Cb(Ω) the sequence Anf converges pointwise to the function 1{∞} which is not continuous. This shows that in Proposition 3.2.9 we cannot expect better convergence than with respect to σ(X,Y0). On the other hand, this example also shows that even if both X and Y have ergodic decompositions, it can depend on the average scheme how strong the convergence to the ergodic projection is.

Let us also note that the average scheme An has the e-property, whereas A˜n does not have the e-property. For the e-property, the only point of interest is the point ∞ as all other points of Ω are isolated. In this situation Cb(Ω) = Lipb(Ω, d). Given f ∈ Cb(Ω) and ε > 0, we ﬁnd n0 such that |f(n) − f(∞)| ≤ ε for all n ≥ n0. Consequently, we also have

|S0kf(n) − S0kf(∞)| = |f(n + k) − f(∞)| ≤ ε

44 3.4. Counterexamples

for all n ≥ n0 and all k ≥ 0. Thus k−1 1 X |A0 f(n) − A0 f(∞)| ≤ |f(n + j) − f(∞)| ≤ ε k k k j=0 N 0 N for all k ∈ and all n ≥ n0, i.e. {Akf : k ∈ } is equicontinuous. On the other hand, Añ cannot have the e-property, since in this case it would follow from ˜0 0 Theorem 3.3.9 that Anf → P f pointwise, which was seen to be wrong above. We have seen in Proposition 3.2.9 (a) that if fix(S ) and fix(S 0) separate each other and are finite dimensional, then both X and Y can be decomposed as in (iv) of Theorem 3.2.4. Our last example shows that this is not true for infinite dimensional fixed spaces.

Example 3.4.3. In the following we construct a positive, contractive and σc- continuous operator S on the norming dual pair (M (Ω),Cb(Ω)) such that, for n S := {S : n ∈ N0}, we have M (Ω) 6= ﬁx(S ) ⊕ rgσc (I − S ) while the ﬁxed spaces of S and S 0 separate each other.

Figure 3.1: Transformation of Ω by the action of ϕ

For n ∈ N let Kn := {0, . . . , n} × {1/n} and K0 := N0 × {0}. On the set Ω := S K endowed with the topology inherited from R2, we consider the n∈N0 n continuous mapping ϕ :Ω → Ω, given by ( ((k + 1), 1/n) n ∈ N, k ∈ {0, . . . , n − 1} ϕ((k, 1/n)) := (0, 1/n) n ∈ N, k = n

45 3. Mean ergodic theorems on norming dual pairs

and ϕ(k, 0) := (k + 1, 0) for all k ∈ N. Thus on each Kn the map ϕ shifts to the 1 1 right and for n 6= 0 the point (n, n ) is mapped to (0, n ), see Figure 3.1. 0 0 Let S denote the induced Koopman operator on Cb(Ω), defined as S f := f ◦ϕ. It is easy to see that ∞ 0 0 X fix(S ) = fix(S ) = an1K : lim an = a0 . n n→∞ n=0 On the other hand, ∞ X 1 fix(S ) = fix(S) = anζn :(an) ∈ ` , n=1

where ζn denotes counting measure on Kn with the normalization ζn(Kn) = 1. It thus follows that the ﬁxed spaces separate each other. We now show that

σc δ(0,0) 6∈ ﬁx(S ) ⊕ span rg(I − S ).

1 Aiming for a contradiction, let us assume that there exists a sequence (an) ∈ ` and a net (µα)α ⊂ span rg(I − S ), σc-converging to µ ∈ M (Ω), such that

∞ X δ(0,0) = anζn + µ. n=1 1 Since Kn is a ﬁxed point of S and µα belongs to span rg(I − S ), we have 1 N 1 h Kn , µαi = 0 for all n ∈ and all α. Thus also h Kn , µi = 0. It follows that 1 0 = h Kn , δ(0,0)i = anζn(Kn) + µ(Kn) = an

0 for all n ∈ N and hence δ(0,0) = µ. Since 1Ω ∈ ﬁx(S ), the contradiction

1 = h1Ω, δ(0,0)i = limh1Ω, µαi = 0 α follows.

46 CHAPTER 4 Kernel and Harris operators

In this chapter we introduce and characterize kernel and Harris operators and study their peripheral point spectrum. On Lp-spaces an operator of the form Z (T f)(x) = h(x, y)f(y) dy for some measurable function h is called a kernel (or integral) operator. Taking a more abstract point of view, it turns out that these operators are precisely the order continuous ones of the band generated by the finite rank operators. Since the latter makes sense in arbitrary Banach lattices, this gives rise to a generalization of this notion. If T is a regular operator between the Banach lattices E and F , where F is assumed to be order complete, we say that T is a kernel operator if it belongs to the band generated by the finite rank operators. If at least some power of T dominates a non-trivial kernel operator, then T is called a Harris operator. After recalling these definitions we provide some basic properties of kernel and Harris operators in Section 4.1. A useful characterization of kernel operators by a continuity condition is due to Bukhvalov. He proved that a linear operator on an Lp-space is a kernel operator if and only if it maps order bounded norm convergent sequences to almost everywhere convergent ones. A characterization of general kernel operators in the spirit of Bukhvalov’s theorem was provided by Grobler and van Eldik in [35]. They replaced Bukhvalov’s continuity condition with the so-called star–order continuity. In Section 4.2 we present an alternative proof of this result under different conditions on the spaces.

47 4. Kernel and Harris operators

Afterwards, in Section 4.3, we study the peripheral point spectrum of a semigroup of Harris operators. Davies proved in [19] that σp(A) ∩ iR ⊆ {0} if A is the generator of a positive and contractive strongly continuous semigroup on `p for some 1 ≤ p < ∞. This result was generalized by Keicher in [39] to generators of positive and bounded semigroups on atomic Banach lattices with order continuous norm. A further generalization to (w)-solvable semigroups on super-atomic Banach lattices was found by Wolﬀ [62]. Independently, Greiner proved in [33, Thm 3.2 and Kor 3.11] that σp(A)∩iR ⊆ {0} for the generator A of a positive and contractive semigroup on Lp(Ω, µ) for 1 ≤ p < ∞ if the semigroup contains a kernel operator. See also [9, Thm 3.1] for this result and [9, Thm 3.5] for a generalization to Abel bounded semigroups. Greiner’s theorem includes in particular Davies’ result since every regular operator on an order complete atomic Banach lattice is a kernel operator by Lemma 4.1.5. Under the assumption that the semigroup under consideration is irreducible, we generalize Greiner’s result in two respects. First, we consider general Banach lattices with order continuous norm instead of Lp-spaces and secondly, we merely assume that one operator of the semigroup dominates a non-trivial compact or kernel operator. All results of Section 4.3 have been published in [26].

4.1 Deﬁnition and basic properties

Let (Ω1, Σ1, µ1) and (Ω2, Σ2, µ2) be σ-ﬁnite measure spaces and (Ω, Σ, µ) be their product space. For p, q ∈ [1, ∞], we call a linear operator

p q T : L (Ω1, Σ1, µ1) → L (Ω2, Σ2, µ2)

an integral operator if there exists a measurable map h: Ω → R such that for p µ2-almost every x ∈ Ω2 and every f ∈ L (Ω1, Σ1, µ1), the function f( · )h( · , y) is 1 in L (Ω1, Σ1, µ1) and Z (T f)(x) = f(x)h(x, y) dy. (4.1.1) Ω1

If, in addition, the operator T is regular, then also |T | is an integral operator and given by Z (T f)(y) = f(x)|h(x, y)| dx, Ω1 see [49, Thm 3.3.5]. This shows that the regular integral operators form a sublattice of the regular operators. Moreover, they are even a band, which is a consequence of Theorem 4.1.2 below.

48 4.1. Deﬁnition and basic properties

Deﬁnition 4.1.1. Let E and F be Banach lattices such that F is order complete. We denote by E∗ ⊗ F the space of all ﬁnite rank operators from E to F . The elements of (E∗ ⊗ F )⊥⊥, the band generated by E∗ ⊗ F in the order complete vector lattice L r(E,F ), are called kernel operators. The following theorem asserts that for order continuous operators on Lp- spaces the notions of integral and kernel operators coincide. Let us remark that ∗ ⊥⊥ ∗ ⊥⊥ (Eoc ⊗ F ) are precisely the order continuous elements of (E ⊗ F ) as shown ∗ ∗ by Arendt in [6, Lem 1.21], where Eoc is the order continuous dual of E .

Theorem 4.1.2 ([55, Prop IV 9.8]). Let (Ω1, Σ1, µ1) and (Ω2, Σ2, µ2) be σ-ﬁnite p q measure spaces, E := L (Ω1, Σ1, µ1) and F := L (Ω2, Σ2, µ2) for some p, q ∈ [1, ∞]. A regular operator T : E → F is an integral operator, i.e. given by (4.1.1) for a measurable function h, if and only if T is an order continuous kernel operator, ∗ ⊥⊥ i.e. T ∈ (Eoc ⊗ F ) . The following Lemma is part of [49, Lem 4.2.7].

Lemma 4.1.3. Let E and F be Banach lattices and assume F to be order complete. For weak units z of F and ϕ of E∗ let A := {ϕ ⊗ z}⊥ and B := A⊥. Then A = (E∗ ⊗ F )⊥ and B = (E∗ ⊗ F )⊥⊥.

Proof. Obviously B ⊂ (E∗ ⊗ F )⊥⊥. In order to prove the converse inclusion, we check that B contains every positive operator of rank one. If y ∈ F+ and ∗ τ ∈ [0, ϕ] ⊂ E+, then it follows from

sup{τ ⊗ (y ∧ nz): n ∈ N} = τ ⊗ y

∗ that τ ⊗ y ∈ B. For y ∈ F+ and τ ∈ E+ we now conclude from

sup{(τ ∧ nϕ) ⊗ y : n ∈ N} = τ ⊗ y that τ ⊗ y ∈ B. Therefore, the ideal B contains every ﬁnite rank operator and hence (E∗ ⊗ F )⊥⊥ ⊂ B. Since every band in the order complete vector lattice L r(E,F ) is a projection band, it follows from

L r(E,F ) = A ⊕ B = (E∗ ⊗ F )⊥ ⊕ (E∗ ⊗ F )⊥⊥ that A = (E∗ ⊗ F )⊥.

Next, we show that on an atomic order complete Banach lattice, e.g. on `p for p ∈ [1, ∞], every order continuous regular operator is a kernel operator. We start with recalling the deﬁnitions of an atom and an atomic space.

49 4. Kernel and Harris operators

Deﬁnition 4.1.4. Let E be a Banach lattice. A vector a ∈ E+ is called an atom if the principal ideal generated by a is one-dimensional. Let

A := {a ∈ E+ : a is an atom and kak = 1}.

The space E is called atomic if A ⊥⊥ = E and diﬀuse if A = ∅. It is easy to verify that in an order complete atomic vector lattice E every element x ∈ E+ is the supremum of a directed set B ⊂ [0, x] where B ⊂ span A . Lemma 4.1.5. Let E and F be order complete Banach lattices and E be atomic. r ∗ ⊥⊥ Then every operator T ∈ Loc(E,F ) belongs to (Eoc ⊗ F ) . Proof. Let T ∈ L r(E,F ) be order continuous and positive and deﬁne

A := {a ∈ E+ : a is an atom and kak = 1}.

∗ For each a ∈ A let ϕa ∈ Eoc denote the order continuous positive functional given by hϕa, ai = 1 and hϕa, a˜i = 0 for alla ˜ ∈ A \{a}. Let N T := {(ϕa1 ⊗ T a1) + ··· + (ϕan ⊗ T an): n ∈ , a1, . . . , an ∈ A },

∗ which is an upwards directed subset of Eoc ⊗ E. Then T is an upper bound of T and (sup T )x = sup{Rx : R ∈ T } = T x

for every x ∈ span A . Let x ∈ E+ and xα ∈ span A an upwards directed net with sup xα = x. As T and all R ∈ T are order continuous, we conclude that

sup{Rx : R ∈ T } = sup{sup Rxα : R ∈ T } α

= sup sup{Rxα : R ∈ T } α

= sup T xα = T x. α

∗ ⊥⊥ This shows that T = sup T ∈ (Eoc ⊗ E) . Deﬁnition 4.1.6. Let E and F be Banach lattices such that F is order complete. An operator T ∈ L r(E,F ) is called Harris, if T n 6∈ (E∗ ⊗ F )⊥ for some n ∈ N. Note that a positive operator is a Harris operator if and only if some power of it dominates a non-trivial kernel operator.

Lemma 4.1.7. Let K ∈ L r(E) be a kernel operator on a Banach lattice E. Then KT,TK ∈ (E∗ ⊗ E)⊥⊥ for every order continuous T ∈ L r(E).

50 4.1. Deﬁnition and basic properties

Proof. We may assume that K,T ≥ 0. Denote by J the ideal generated by E∗ ⊗E in L r(E) and let A := [0,K] ∩ J. Then K = sup A by [49, Prop 1.2.6] and

Kx = sup{Ax : A ∈ A } for every x ∈ E+ since A is upwards directed. Now consider the directed set A T := {AT : A ∈ A }. As every operator of A is dominated by a ﬁnite rank operator, so is every AT , i.e. A T ⊆ J. Now, it follows from

KT x = sup{AT x : A ∈ A }

∗ ⊥⊥ for every x ∈ E+ that KT = sup A T and hence KT ∈ (E ⊗ E) . By a similar argument, using in addition the order continuity of T , we observe that TK ∈ (E∗ ⊗ E)⊥⊥.

Let us denote by

s fix (S ) := {x ∈ E+ : T x ≥ x for all T ∈ S } the set of all super fixed points of a semigroup S . For a positive operator T ∈ L (E), we write fixs(T ) for the super fixed points of the time-discrete n semigroup {T : n ∈ N0}. Lemma 4.1.8. Let S be a positive semigroup on a Banach lattice E. Each of the following conditions asserts that fixs(S ) = fix(S ). In this case fix(S ) is a sublattice of E.

(a) S is contractive and the norm on E is strictly monotone, i.e. kxk < kyk for all x, y ∈ E+ where x < y. (b) ﬁx(S ∗) contains a strictly positive element.

Proof. Easily, fixs(S ) = fix(S ) follows from both, condition (a) and (b). Since for each x ∈ fix(S ) and T ∈ S we have that |x| = |T x| ≤ T |x|, |x| ∈ fixs(S ). Thus, fix(S ) is a sublattice of E if fixs(S ) = fix(S ).

Lemma 4.1.9. Let T ∈ L (E)+ be a power-bounded Harris operator on a Banach lattice E with order continuous norm. Assume that ﬁxs(T ) = ﬁx(T ) contains a n quasi-interior point of E+. Then T ≥ C > 0 for some n ∈ N and a compact operator C ∈ L r(E). Proof. After replacing T with a suitable power T m, we may assume that T ≥ K > 0 for some kernel operator K ∈ L (E)+. Let S := T − K. Since the kernel operators form an algebra ideal by Lemma 4.1.7,

n n n n ∗ ⊥⊥ Kn := T − S = (K + S) − S ∈ (E ⊗ E)

51 4. Kernel and Harris operators

for all n ∈ N. Let z ∈ fix(T ) be a quasi-interior point of E+. It follows from z = T z = Kz + Sz > Sz that Sn+1z ≤ Snz for all n ∈ N. By the order continuity of the n norm, u := limn→∞ S z ∈ fix(S) exists and satisfies 0 ≤ u < z. We infer from T u = Ku + Su ≥ u that u ∈ fixs(T ) = fix(T ). Hence, v := z − u > 0 is a fixed n point of T and lim S v = 0. As the operators Kn are uniformly bounded, it follows from n n Knv = T v − S v → v (n → ∞) 2 2 N that lim Knv = v. In particular, Km =6 0 for some m ∈ . Let J be the ideal generated by E∗ ⊗ E in L r(E). Then

Km = sup([0,Km] ∩ J)

by [49, Prop 1.2.6]. Thus, there exists a sequence (Ck) ⊆ [0,Km] ∩ J such that lim Ckx = Kmx for x ∈ {v, Kmv}. Since every Ck is dominated by a ﬁnite rank 2 N operator, Ck is compact for every k ∈ by [49, Cor 3.7.15]. By the boundedness of the sequence (Ck), we conclude from

2 2 kCk v − Kmvk ≤ kCk(Ckv − Kmv)k + k(Ck − Km)Kmvk

2 2 2 2 2m that lim Ck v = Kmv =6 0. Therefore, 0 < Ck ≤ Km ≤ T for a suitable k ∈ N.

Arendt showed in [6, Kor 1.26] that compact operators are disjoint from lattice isomorphisms on a diﬀuse and order complete Banach lattice. See also [37, Sec 4.4] for similar results. For the sake of completeness, we give a proof of this fact under the assumption that the norm is order continuous.

Theorem 4.1.10. Let V,K : E → E be positive operators on a diﬀuse Banach lattice E with order continuous norm. If V is a lattice isomorphism and K is compact, then V ∧ K = 0.

Proof. First we show that S := I ∧ K = 0. Aiming for a contradiction, we assume that Sx > 0 for some x ∈ E+. We pick some c > 0 such that w := cSx − x is not negative, i.e. it has a non-trivial positive part. Denote by P the band projection onto {w+}⊥⊥, the band generated by w+. Then

0 < w+ = P w = P (cSx − x) = cP Sx − P x.

It follows easily from S ≤ I that S is an orthomorphism, i.e. it commutes with every band projection. Thus, SP x > 0 and hence P x > 0. Since E is diﬀuse, there exists a sequence (xn) ⊆ E of pairwise disjoint elements satisfying 0 < xn < P x. ⊥⊥ Let Pn denote the band projection onto {xn} and deﬁne un := Pnx/kPnxk. As

52 4.2. Characterization by star–order continuity the orthomorphism S is dominated by the compact operator K, S itself is compact by [4, Thm 16.21]. Hence, after passing to a subsequence, (Sun) converges to some y ∈ E. By the order continuity of the norm, the disjoint and order bounded sequence (Pny) converges to zero, see [49, Thm 2.4.2]. Now, it follows from

kPnxkun = Pnx = PnP x ≤ cPnSP x = cPnSx = ckPnxkSun for all n ∈ N that

Pny = lim PnSum ≤ lim PnScSum = cSPny m→∞ m→∞ and hence

2 kunk ≤ ckPnSunk ≤ ckPnSun − Pnyk + ckPnyk ≤ ckSun − yk + c kSPnyk. for all n ∈ N. The right-hand side tends to zero, which is a contradiction to kunk = 1. Hence, S = I ∧ K = 0. To complete the proof we use that the mapping T 7→ TV is a lattice homomorphism on L r(E), see [4, Thm 7.4]. This implies that V ∧ K = (I ∧ KV −1)V vanishes since KV −1 is compact.

4.2 Characterization by star–order continuity

A celebrated theorem by Bukhvalov[16] characterizes integral operators by a continuity condition in the following way, see [65, Thm 96.5].

Theorem 4.2.1 (Bukhvalov). Let (Ω1, Σ1, µ1) and (Ω2, Σ2, µ2) be σ-ﬁnite mea- p sure spaces and let p, q ∈ [1, ∞]. A linear operator T from L (Ω1, Σ1, µ1) to q L (Ω2, Σ2, µ2) is an integral operator if and only if lim T fn = 0 µ2-almost ev- p erywhere for every order bounded sequence (fn) ⊂ L (Ω1, Σ1, µ1) such that every subsequence of (fn) contains a subsequence that converges to 0 µ1-almost everywhere.

Let us ﬁrst compare the condition from Bukhvalov’s theorem that every subsequence contains a subsequence that converges almost everywhere with diﬀerent kinds of convergence in Lp-spaces.

Remark 4.2.2. Let (Ω, Σ, µ) be a σ-ﬁnite measure space, 1 ≤ p ≤ ∞ and (fn) ⊂ Lp(Ω, Σ, µ). It is well-known that the following assertions are equivalent (see e.g. [15, Thm 2.2.3 and Thm 2.2.5]).

(i) Every subsequence of (fn) contains a subsequence that converges to 0 almost everywhere.

53 4. Kernel and Harris operators

(ii) The sequence (fn) converges to 0 in measure on every subset of Ω of ﬁnite measure.

p If p < ∞ and the sequence (fn) is order bounded in L , then the dominated convergence theorem asserts that assertion (i) is equivalent to the following.

p (iii) The sequence (fn) converges to 0 in the norm of L . In this section we present a generalization of Bukhvalov’s theorem to operators on Banach lattices. We characterize kernel operators in the space all of regular operators by a certain kind of order continuity. Similar results have been obtained by Nakano[51] as well as Grobler and van Eldik[35]. In order to generalize Bukhvalov’s theorem, we need to replace almost everywhere convergence with order convergence in its formulation. This results in the notion of star convergence.

Deﬁnition 4.2.3. We say that a sequence (xn) in a vector lattice E star converges to zero, in symbols ?-lim xn = 0, if every subsequence of (xn) contains a subsequence that order converges to zero. Remark 4.2.4. Recall that, in the case where E is an order complete Lp-space, a sequence of functions (fn) ⊂ E converges to zero almost everywhere if and only if |fn| ∧ g converges to zero in order for every g ∈ E+. Indeed, if (fn) converges to zero almost everywhere, then for each g ∈ E+, the sequence gk := supn≥k|fn| ∧ g is monotonically decreasing with inf gk = 0, which shows that o-lim|fn| ∧ g = 0. N If, conversely, for each j ∈ , the sequence gk := supn≥k|fn| ∧ |fj| order converges to zero, then lim fn = 0 almost everywhere on the carrier of each fj. Thus, lim fn = 0 almost everywhere. In particular, an order bounded sequence (fn) converges almost everywhere if and only if it converges in order. Thus, star convergence is a generalization of the condition from Bukhvalov’s theorem to arbitrary vector lattices. A vector lattice E is said to have the countable sup property if for each set A ⊂ E whose supremum exists in E, there is a countable subset C ⊂ S such that sup C = sup A. It is easy to verify that every Banach lattice with order continuous norm has the countable sup property. Furthermore, if a vector lattice admits a strictly positive linear functional ϕ ∈ E∗, i.e. hϕ, xi > 0 for all vectors x > 0, then the space has the countable sup property, see [3, Thm 8.22]. If a vector lattice E has the countable sup property, then clearly every countably order continuous operator on E is order continuous. A set G ⊂ E∗ is said to separate the points of a set F ⊂ E if for each x ∈ F \{0} there exists ϕ ∈ G such that hϕ, xi= 6 0. The next Lemma compares star convergence with weak convergence and convergence in norm.

54 4.2. Characterization by star–order continuity

Lemma 4.2.5. Let E be an order complete Banach lattice, (xn) ⊂ E+ and ∗ ∗ G ⊂ Ecoc ∩ E+. Consider the following assertions.

(i) limkxnk = 0

(ii) ?-lim xn = 0

(iii) limhϕ, xni = 0 for every ϕ ∈ G.

Then (i) ⇒ (ii) ⇒ (iii). If G separates the points of E+, E has the countable sup property and (xn) is order bounded, then (iii) ⇒ (ii). If the norm on E is order continuous, then (ii) ⇒ (i).

Proof. It was shown in [65, Thm 100.6] that every norm convergent sequence has a subsequence that converges in order. Thus, (i) implies (ii). Now assume that ?-lim xn = 0. By assumption there exists a subsequence (xnk ) of (xn) such ∗ that o-lim xnk = 0. In particular, limhϕ, xnk i = 0 for all ϕ ∈ Ecoc. Thus, (xnk ) ∗ converges to zero in the locally convex topology σ(E,Ecoc). Since we can apply this argument also to every subsequence of (xn), it follows that limhϕ, xni = 0 for ∗ all ϕ ∈ Ecoc. This shows that (ii) implies (iii). Now assume that (iii) holds, that E has the countable sup property and that G separates the points of E+. Moreover, we assume that 0 ≤ xn ≤ x for some x ∈ E+. For every ϕ ∈ G we denote by N(ϕ) = {z ∈ E : hϕ, |z|i = 0} the null ⊥ ideal of ϕ and by Pϕ the band projection onto the carrier C(ϕ) := N(ϕ) of ϕ. ∗ ∗ Note that the countable sup property implies that every Ecoc = Eoc. Therefore, N(ϕ) is a band and hence C(ϕ)⊥ = N(ϕ) for every ϕ ∈ G. Since G separates the points of E+, this implies that

sup{Pϕx : ϕ ∈ G} = x.

Hence, by the countable sup property, we ﬁnd countably many functionals (ϕm) ⊂

G such that x = supm∈N Pϕm x. Let m ∈ N and consider the seminorm kxkm := hϕm, |x|i. We denote by Fm the completion of E/N(ϕm) with respect to k · km. Since limkxnkm = 0, it follows (m) from [65, Thm 100.6] that there exists a subsequence (xnk ) of (xn) that order (m) converges to 0 in Fm, i.e. ym := infn∈N supk≥n xnk satisﬁes hϕ, ymi = 0. So far, for each m ∈ N we have constructed a subsequence of (xn). Choosing a diagonal sequence, denoted by (xnk ), we obtain that y := infn∈N supk≥n xnk satisﬁes hϕm, yi = 0 for all m ∈ N. Since 0 ≤ y ≤ x, we have that y = supm∈N Pϕm y = 0. Thus, o-lim xnk = 0. In particular, (xnk ) star converges to 0. Hence, we proved that (iii) implies (ii). If ?-lim xn = 0 and the norm on E is order continuous, then every subsequence of (xn) has a subsequence that norm converges to 0, thus limkxnk = 0.

55 4. Kernel and Harris operators

Definition 4.2.6. Let E and F be Banach lattices. A linear operator T : E → F is called star–order continuous (or s-o-continuous), if o-lim T xn = 0 for every order bounded sequence (xn) ⊂ E+ that star converges to 0. It is clear that star–order continuous operators are countably order continuous. In what follows, we prove that they are precisely the countably order continuous kernel operators. As a first step we establish that star–order continuous operators form a band. This was first proven by Grobler and van Eldik in [35, Prop 2.3] and independently by Vietsch in [61, Thm 6.1]. Proposition 4.2.7. Let E and F be order complete Banach lattices. Then

r L?o := {T ∈ L (E,F ): T is star–order continuous}

is a band in L r(E,F ).

r Proof. It is easy to check that L?o is a subspace of L (E,F ). We prove now that L?o is also an ideal by following the final steps of the proof of [65, Thm 96.5]. Let T ∈ L?o, x ∈ E+ and (xn) ⊂ [0, x] such that ?-lim xn = 0. Define yn := x − xn and fix y ∈ [0, x]. Since the lattice operations are continuous with respect to order convergence, we have ?-lim(y − y ∧ yn) = 0. Thus, by the s-o-continuity of T we obtain that o-lim T (y ∧ yn) = T y. Hence,

+ T y = o-lim T (yn ∧ y) ≤ lim inf T yn n→∞ n→∞ and consequently + + T x = sup T y ≤ lim inf T yn. 0≤y≤x n→∞

Conversely, it follows from 0 ≤ yn ≤ x that

+ + lim sup T yn ≤ T x. n→∞

+ + This shows that o-lim T yn = T x and consequently

+ + + o-lim T xn = o-lim T x − T yn = 0.

+ r This proves that T ∈ L?o and hence that L?o is a sublattice of L (E,F ). r Let T ∈ L (E,F ) and S ∈ L?o such that |T | ≤ |S|. Then |S| ∈ L?o by the ﬁrst part of this proof. Thus, for every order bounded sequence (xn) ⊂ E+ with ?-lim xn = 0 we have that

o-lim|T xn| ≤ o-lim|S|xn = 0.

r This proves that T ∈ L?o. Thus, L?o is an ideal in L (E,F ).

56 4.2. Characterization by star–order continuity

In order to prove that L?o is closed under taking suprema, we follow the arguments of the proof of [49, Prop 1.3.9]. Let A ⊂ L?o such that S := sup A exists in L r(E,F ). We may assume that A is upwards directed and consists of positive operators. Then Sx = supT ∈A T x for all x ∈ E+. Let x ∈ E+ and (xn) ⊂ [0, x] such that ?-lim xn = 0. Let y := lim sup Sxn. In order to prove that S ∈ L?o, we have to show that y = 0. Fix T ∈ A . Since T (x − xn) ≤ S(x − xn) for all n ∈ N we obtain that Sxn + T x ≤ Sx + T xn. Therefore,

y + T x ≤ sup Sxn + T x ≤ sup(Sxn + T x) ≤ sup(Sx + T xn) = Sx + sup T xn n≥k n≥k n≥k n≥k for all k ∈ N. For k → ∞, we obtain that y + T x ≤ Sx. Since T ∈ A was arbitrary, it follows that y ≤ 0 which completes the proof. We recall that a topological space is called extremely disconnected (or Stonian) if the closure of every open set is open. Extremely disconnected spaces are precisely those for which the vector lattice of continuous functions is order complete, see [55, Prop II 7.7]. Theorem 4.2.8. For a compact space K, the M-space C(K) is order complete if and only if K is extremely disconnected. In the following Lemma we note a surprising property of a regular Borel measure on a compact and extremely disconnected space K. Lemma 4.2.9. Let K be an extremely disconnected compact space and λ be a positive regular Borel measure on K. Then λ(B) = λ(B) for every Borel set B ⊂ K. Proof. Let us ﬁrst consider an open set B ⊂ K. Deﬁne

F := {f ∈ C(K) : 0 ≤ f ≤ 1B}. 1 Since C(K) is order complete, there exists g := sup F . We show that g = B. 1 Obviously, B is an upper bound of F . Assume that g(x) < 1 for some x ∈ B. Then there exists y ∈ B such that g(y) < 1. By Urysohn’s lemma, we can construct a function h ∈ C(K) satisfying h(y) = 1 and 0 ≤ h ≤ 1B. Then h ∈ F 1 but h 6≤ g. This is impossible and it follows that g = B. Now we conclude from [15, Lem 7.2.6] that λ(B) = hλ, gi = sup{hλ, fi : f ∈ F } ≤ λ(B). This proves that λ(B) = λ(B). Now consider an arbitrary Borel set B ⊂ K and assume that λ(B) > λ(B). By the regularity of λ we ﬁnd an open set U ⊃ B such that λ(U) < λ(B). But by the ﬁrst part of the proof we know that λ(U) = λ(U) ≥ λ(B). This is impossible and we conclude that λ(B) = λ(B).

57 4. Kernel and Harris operators

In the proof of the following Proposition we make use Kakutani’s representation theorem for M-spaces, see [55, Thm II 7.4]. Theorem 4.2.10 (Kakutani). Let E be a M-space with unit. Then M is isometric isomorphic to C(K) for a compact space K. Proposition 4.2.11. Let E and F be order complete Banach lattices and T ∈ ∗ ∗ L (E,F ) be a positive operator. Let x ∈ E+ and ϕ ∈ Eoc ∩ E+ such that T ∧ (ϕ ⊗ T x) = 0. Then there exists a sequence of band projections Pn such that limhϕ, (I − Pn)xi = 0 and lim inf TPnx = 0. Proof. We may assume that T x > 0 and hϕ, xi > 0, otherwise the assertion follows with the constant sequences Pn = I and Pn = 0, respectively. Note that the order continuous functional ϕ is strictly positive on its carrier C(ϕ) = N(ϕ)⊥. Therefore, we may assume that ϕ is strictly positive on the principal ideal spanned by x, otherwise we replace x with the projection of x onto C(ϕ). By [49, Prop 1.2.13] the ideals Ex and FT x are order complete M-spaces with respect to the order unit norm. By Kakutani’s theorem (Theorem 4.2.10) these M-spaces are isometric isomorphic to C(K) and C(L) for certain extremely disconnected compact spaces K and L. In what follows, we consider T as a positive mapping from C(K) to C(L) satisfying T 1 = 1 and ϕ as a strictly positive regular measure λ on the Borel σ-algebra B(K). By C (K) we denote the set of all closed and open (clopen) subsets of K. Since (T ∧ (ϕ ⊗ 1))1 = 0, it follows from [5, Thm 1.50] that

0 = inf{T 1A + ϕ(K \ A)1 : A ∈ C (K)} (4.2.1) = inf{T 1A + ϕ(K \ A)1 : A ∈ C (K),A 6= K}. For each m ∈ N deﬁne 1 Gm := T 1A : A ∈ C (K), 0 < λ(K \ A) ≤ m

and let h := inf Gm. It follows from (4.2.1) that 1 1 0 = (T ∧ (ϕ ⊗ 1))1 ≥ h ∧ 1 ≥ h m m and hence h = 0. Thus, by Urysohn’s lemma, the open set

[ 1 U := g < m m g∈Gm

is dense in K and hence, by Lemma 4.2.9, λ(Um) = λ(K). Now let ε > 0. For −m each m ∈ N we ﬁnd a compact set Km ⊂ Um of measure λ(Km) ≥ λ(K) − ε2

58 4.2. Characterization by star–order continuity

by the regularity of λ. By compactness, there are ﬁnitely many g1,m, . . . , gnm,m such that 1 1 K ⊂ g < ∪ · · · ∪ g < . m 1,m m nm,m m 1 Since each gj,m belongs to Gm, we have that gj,m = T Aj,m for a Aj,m ∈ C (K) 1 such that λ(K \ Aj,m) ≤ m . Denote by (An) the sequence of all these sets Aj,m, i.e.

(An) := A1,1,...,An1,1,A1,2,...,An2,2,...,A1,3,...,An3,3,... T Then for Dε := m∈N Km we have inf T 1A (t) = 0 n≥k n

N 1 for all t ∈ Dε and k ∈ , i.e. lim inf T An = 0 on Dε. 1 k We proved that for ε := k there exists a sequence (An)n∈N ⊂ C (K) such that k lim λ(K \ A ) = 0 and lim inf T 1 k = 0 on D . Deﬁne n→∞ n n→∞ An 1/k

1 A := Ak : λ(K \ Ak ) ≤ , k, n ∈ N . n n k

Then A is a countable set that contains all but ﬁnitely many elements of each k sequence (An)n∈N. We can now easily order the elements of A increasing by their measure λ since for every δ > 0 there are only ﬁnitely many elements in A of measure less then δ and none of measure λ(K). Let us denote this enumeration of A by (Bn)n∈N. Then λ(Bn) converges monotonically increasing to λ(K) and 1 lim inf T Bn = 0 on D := ∪k∈ND1/k. Since 1 λ(D) ≥ lim λ(D ) = lim λ(K) − = λ(K), 1/k k 1 D is dense in K by Lemma 4.2.9. Thus lim inf T Bn = 0 on K and the assertion follows if we choose Pn as the projection onto the principal band generated by 1 Bn .

Now we turn to the proof of the announced main result of this section, a ∗ diﬀerent version of [35, Thm 5.6] where Foc is assumed to separate the points of F .

Theorem 4.2.12. Let E and F be order complete Banach lattices where E has ∗ the countable sup property. Assume that Eoc separates the points of E+. Then

r ∗ ⊥⊥ L?o := {T ∈ L (E,F ): T is star–order continuous } = (Eoc ⊗ F ) .

59 4. Kernel and Harris operators

Proof. Since every order continuous ﬁnite rank operator is s-o-continuous and L?o is a band by Proposition 4.2.7, we have that

∗ ⊥⊥ (Eoc ⊗ F ) ⊂ L?o.

∗ ⊥ In order to prove the converse inclusion, let T ∈ (Eoc ⊗ F )+ be s-o-continuous. ∗ ∗ Fix ϕ ∈ Eoc ∩ E+ and let x ∈ C(ϕ)+. Since T ∧ (ϕ ⊗ T x) = 0, we ﬁnd a sequence 0 ≤ yn ≤ x by Proposition 4.2.11 such that limhϕ, x−yni = 0 and lim inf T yn = 0. Since {ϕ} separates the points of C(ϕ)+ and C(ϕ) has the countable sup property, Lemma 4.2.5 yields that ?-lim(x − yn) = 0 and therefore o-lim T (x − yn) = 0 as T is s-o-continuous. Thus, 0 = lim inf T yn = o-lim T yn = T x and, as x ∈ C(ϕ)+ was ∗ ∗ arbitrary, we conclude that T = 0 on C(ϕ). Since Eoc ∩ E+ separates the points ∗ ⊥ of E+, it follows that T = 0 is the only s-o-continuous operator in (Eoc ⊗F )+. Taking Theorem 4.1.2 and Remark 4.2.4 into account, Bukhvalov’s theorem as stated in Theorem 4.2.1 follows immediately from Theorem 4.2.12. Moreover, Theorem 4.2.12 enables us to show that, for a given semigroup S of kernel operators on a Banach lattice E, we can decompose ﬁx(S )⊥⊥ in at most countably many S -invariant disjoint bands such that the restriction of S to each band is irreducible.

Theorem 4.2.13. Let S be a positive semigroup on a Banach lattice E with order continuous norm. Assume that fixs(S ) = fix(S ) and that some operator T0 ∈ S is a kernel operator. Then for every z ∈ fix(S )+ there exist at most P countably many disjoint band projections (Pn) with z = Pnz such that PnE is S -invariant and the restriction of S to PnE is irreducible for every n ∈ N.

⊥⊥ Proof. Let us fix z ∈ fix(S )+. Since the band {z} is an M-space with unit z (see the Corollary to [55, Prop II 7.2]), we may identify it with some C(K) by Kakutani’s theorem (Theorem 4.2.10) and thus consider S as a contractive semigroup on C(K). The order continuity of the norm on E implies that {z}⊥⊥ =∼ C(K) is order complete and has the countable sup property. We denote by C (K) the set of all closed and open (clopen) subsets of K and observe that T 1B ≤ 1 for every B ∈ C (K) and every T ∈ S . Let us call a set B ∈ C (K) invariant if T 1B ≤ 1B for all T ∈ S and irreducible if for every invariant clopen subset A ⊂ B we have A = ∅ or A = B. With this identification and notation, we have to find at most countable many disjoint, invariant and irreducible sets (Dn) ⊂ C (K) such that K = D1 ∪ D2 ∪ ... . First, we show that a set B ∈ C (K) is invariant if and only if K \B is invariant if and only if both 1B and 1K\B belong to fix(S ). Let B ∈ C (K) be invariant. Then, for every T ∈ S ,

T 1K\B = T 1 − T 1B ≥ 1 − 1B = 1K\B.

60 4.3. Triviality of the peripheral point spectrum

s Since fix (S ) = fix(S ), 1K\B is a fixed point of T and so is 1B. Next, we prove the existence of an irreducible non-empty clopen set. Aiming for a contradiction, we assume that ∅ is the only irreducible clopen set. For n ∈ N define 1 An := 1A : A ∈ C (K) is invariant and k1AkE ≤ . n

By the order completeness of C(K) the supremum of An exists and it is easy to 1 check that sup An = Bn for some Bn ∈ C (K). Then Bn, hence by the above also K \ Bn, is invariant because every operator T ∈ S is order continuous. Since 1 1 we assumed that ∅ is the only irreducible set and k AkE > n for every clopen invariant subset A ⊂ K \ Bn, it follows that K \ Bn = ∅ and Bn = K. Let Dn be a maximal disjoint system in An. By the countable sup property of 1 1 1 C(K), there exists a countable subset ( Ak,n )k∈N ⊂ Dn with supk∈N Ak,n = . 1 N Since the functions { Ak,n : k ∈ } are pairwise disjoint and the norm on E 1 is order continuous, it follows from [49, Thm 2.4.2] that limk→∞k Ak,n kE = 0 for every n ∈ N. By ordering the sets Ak,n decreasing in the norm on E, we obtain a single sequence (An) ⊂ C (K) that contains every set Ak,n and satisﬁes 1 1 limk An kE = 0. In particular, we have that ?-lim An = 0 by Lemma 4.2.5. Now 1 it follows from Theorem 4.2.12 that o-lim T (t0) An = 0 in contradiction to 1 1 1 sup T (t0) An = sup An = n≥k n≥k for all k ∈ N. Thus, there exists an irreducible non-empty set B ∈ C (K). Moreover, the same argument shows that every non-empty invariant clopen set contains a non-empty irreducible clopen set. Let D := {D ∈ C (K): D is irreducible}. Then sup{1D : D ∈ D} = 1B for some B ∈ C (K). Since B and hence K \ B is invariant, it follows that B = K. Thus, by the countable sup property, we ﬁnd a sequence (Dn) ⊂ D with 1 1 sup Dn = as desired.

4.3 Triviality of the peripheral point spectrum

Given a positive, irreducible, bounded and strongly continuous semigroup on a Banach lattice with order continuous norm, we prove that the peripheral point spectrum of its generator is trivial whenever one of its operators dominates a non- n trivial compact or kernel operator. For a time-discrete semigroup {T : n ∈ N0} we show that the point spectrum of some power T k intersects the unit circle at most in 1, which generalizes [34, Thm 3.6]. Similar results are provided for Abel bounded semigroups.

Throughout, let E be a Banach lattice with order continuous norm. By EC we denote the complexiﬁcation of E and by TC : EC → EC the complexiﬁcation

61 4. Kernel and Harris operators

of a linear operator T , see Chapter 1. If S = (T (t))t≥0 is a strongly continuous semigroup on E with generator A, then so is SC := (T (t)C)t≥0 with generator AC. We recall that a strongly continuous semigroup with generator A is said to be Abel bounded if Re λ ≤ 0 for all λ ∈ σ(AC) and supλ>0kλR(λ, AC)k < ∞.A n time-discrete semigroup S = {T : n ∈ N0} is called Abel bounded if |λ| ≤ 1 for −1 all λ ∈ σ(TC) and supλ>1k(λ − 1)(λ − TC) k < ∞. Lemma 4.3.1. Let S be a positive semigroup on E that is strongly continuous or time-discrete. Let z ∈ fixs(S ) \{0} and assume that S is Abel bounded. Then there exists 0 < ϕ ∈ fix(S ∗). If, in addition, S is irreducible, then ϕ is strictly positive and z is a quasi- interior point of E+ with fix(S ) = span{z}.

n Proof. For a time-discrete semigroup S = {T : n ∈ N0}, we first observe that r(T ) = 1. Indeed, the Abel boundedness implies that r(T ) ≤ 1 and since T nz ≥ z > 0 for all n ∈ N we have r(T ) ≥ 1. Now the existence of 0 < ϕ ∈ fix(S ∗) follows from [55, Lem V 4.8]. For a strongly continuous semigroup, we argue as in the proof of [11, Prop ∗ ∗ 4.3.6]. Fix τ ∈ E+ such that hτ, zi = 1 and let ϕ be a σ(E ,E)-cluster point of λR(λ, A)∗τ as λ ↓ 0. Since Z ∞ R(λ, A)x = e−λsT (s)x ds 0 for all x ∈ E, we have R(λ, A) ≥ 0 and λR(λ, A)z ≥ z for all λ > 0. Therefore, ϕ ≥ 0 and hϕ, zi ≥ hτ, zi = 1. For every y ∈ D(A) it follows from the boundedness of λR(λ, A) that lim λR(λ, A)Ay = lim λ(λR(λ, A)y − y) = 0 λ↓0 λ↓0 and hence hAy, ϕi = limhλR(λ, A)Ay, τi = 0. λ↓0 This proves that ϕ ∈ ker A∗ = fix(S ∗). Now assume that S is irreducible. Since the null ideal N(ϕ) := {x ∈ E : hϕ, |x|i = 0} is closed, S -invariant and different from E, N(ϕ) = {0}. Thus, ϕ is strictly positive and therefore fix(S ) = fixs(S ) is a sublattice of E by Lemma 4.1.8. For every x ∈ fix(S ), the principal ideals generated by the fixed points x+ and x− are S -invariant and disjoint. Since S is irreducible, either x+ is a quasi-interior point − of E+ and x = 0 or vice versa. This implies that fix(S ) is totally ordered and hence one-dimensional by [55, Prop II 3.4]. In particular, fix(S ) = span{z}.

62 4.3. Triviality of the peripheral point spectrum

We continue with a consequence of the mean ergodic theorem, Theorem 3.0.8.

Lemma 4.3.2. Let T ∈ L (E)+ be Abel bounded. If there exists a quasi-interior point z ∈ ﬁx(T ) of E+, then T is mean ergodic.

N 1 Pn−1 k Proof. For n ∈ we denote by An := n k=0 T the Ces`aroaverages of T . It is well-known that the Abel boundedness of T implies that M := supn∈NkAnk < ∞, see [22, 1.5]. Hence,

n n T T ≤ 2 ≤ 2kAn+1k ≤ 2M n n + 1

N 1 n for all n ∈ . Now, it follows from lim n T x = 0 for all x in Ez, the principal 1 n ideal generated by z, that lim n T x = 0 for all x ∈ E. Due to the order continuity of the norm on E, for each c > 0 the An-invariant order interval [−cz, cz] is σ(E,E∗)-compact, see [49, Thm 2.4.2]. Thus, the sequence (Anx) has a weak cluster point for every x ∈ Ez and the mean ergodic theorem, Theorem 3.0.8, implies that lim Anx exists for all x ∈ Ez. Consequently, by the uniform boundedness of the operators An and the density of Ez, lim Anx exists for all x ∈ E.

The following theorem is a special version of the famous splitting theorem by Jacobs, de Leeuw and Glicksberg. Here and for the rest of this section, we denote by Lσ(E) the space of all bounded linear operator on E endowed with the weak operator topology.

Theorem 4.3.3 (Jacobs – de Leeuw – Glicksberg). Let S ⊂ L (E) be a positive, bounded and Abelian semigroup and T its closure in Lσ(E). If there exists a quasi-interior point z ∈ ﬁx(S ) of E+, then T contains a positive projection Q commuting with every operator in T such that the closed subspace F := QE includes ﬁx(S ) and QT := {QT : T ∈ T } ⊆ T is a norm bounded group of positive operators with neutral element Q. If, in addition, S is irreducible, then Q is strictly positive, i.e. Qx > 0 for all x > 0.

Proof. It follows from the uniform boundedness principle that the closure of the norm bounded set S in Lσ(E) is again bounded in norm. Moreover, the operators of T are positive and commute since S is Abelian. Let x ∈ Ez, i.e. x ∈ [−cz, cz] for some c > 0. Then {T x : T ∈ S } is a subset of [−cz, cz] and hence relatively weakly compact by the order continuity of the norm, see [49, Thm 2.4.2]. Since Ez is dense in E, S is relatively compact in Lσ(E) by [23, Cor A 5]. Now, it follows from [44, §2 Thm 4.1] that there exists a positive projection Q ∈ T that commutes with every operator in T and QT is a

63 4. Kernel and Harris operators

bounded group with neutral element Q. Since Q is in the Lσ(E)-closure of S , one has that fix(S ) ⊆ fix(Q) ⊆ F . If, in addition, S is irreducible, then Q is strictly positive since the null ideal N(Q) = {y ∈ E : Q|y| = 0} is closed, S -invariant, different from E and therefore equal to {0}.

Remark 4.3.4. Let a ∈ E+ be an atom and T ∈ L (E)+. Then clearly TEa ⊆ ET a. If T is a lattice isomorphism, i.e. it is invertible with a positive inverse, then

−1 ET a = TT ET a ⊆ TET −1T a = TEa = span{T a}.

Hence, every lattice isomorphism on E maps atoms to atoms. The lemma below is a more general formulation of [39, Prop. 3.5], followed by an analogon for time-discrete semigroups.

Lemma 4.3.5. Let T = (T (t))t∈R be a positive and bounded strongly continuous group on E, i.e. T is a uniformly bounded family of positive linear operators on E such that T (0) = I, T (t)T (s) = T (t + s) for all t, s ∈ R and the mappings t 7→ T (t)x are continuous from R to E for every x ∈ E. Then ﬁx(T ) contains every atom of E.

Proof. Since T is a group, every operator T (t) is a lattice isomorphism and hence maps atoms to atoms by Remark 4.3.4. Fix an atom a ∈ E+. Then for every disjoint atom b ∈ E+ one has that |a − b| = a + b and hence ka − bk ≥ kak. Thus, it follows from limt→0 T (t)a = a that T (t)a ∈ Ea for all t ∈ R. Therefore, T (t)a = exp(λt)a for a λ ∈ R and all t ∈ R by the semigroup law. Since the group T is bounded, we conclude that λ = 0 and hence a ∈ ﬁx(T ).

Lemma 4.3.6. Suppose that E is not diﬀuse and let T be an irreducible lattice isomorphism on E such that

sup{kT kk : k ∈ Z} < ∞. (4.3.1)

Then there exists n ∈ N such that T n = I.

Proof. Let a ∈ E+ be an atom. The ideal

J := {x ∈ E : |x| ≤ c(a + T a + ··· + T ma) for some c ≥ 0 and m ∈ N}

is T -invariant and hence dense in E. Since the norm on E is order continuous, J ⊥⊥ equals J , the band generated by J. Thus, for every x > 0 we ﬁnd some n ∈ N0 such that x ∧ T na > 0. In particular, T −1a ∧ T n−1a > 0 for some n ∈ N. As T −1 maps atoms to atoms by Remark 4.3.4, this implies that T n−1a = cT −1a for some

64 4.3. Triviality of the peripheral point spectrum c > 0 and T na = ca. It follows from assumption (4.3.1) that c = 1 and hence k n k T a ∈ ﬁx(T ) for all k ∈ N0. Since every T a is an atom, we conclude that E = J = span{a, T a, . . . , T n−1a} ⊆ ﬁx(T n), which completes the proof. The following theorem is the main result of this section and a generalization of [9, Thm 3.1], where the semigroup is assumed to contain a kernel operator.

Theorem 4.3.7. Let T = (T (t))t≥0 be a positive, irreducible and bounded strongly continuous semigroup with generator A. If there exists t0 > 0 such that T (t0) ≥ K > 0 where K is a compact or a kernel operator, then σp(AC)∩iR ⊆ {0}.

Proof. Let α ∈ R such that iα ∈ σp(AC) ∩ iR. Then T (t)Cz = exp(iαt)z for some z ∈ EC\{0} by the spectral inclusion theorem [23, Thm IV 3.6]. Since 0 < |z| = |T (t)Cz| ≤ T (t)|z| for all t ≥ 0, it follows from Lemma 4.3.1 that there exists a strictly positive ϕ ∈ fix(T ∗) and fix(T ) = span{e} where e := |z| is a quasi-interior point of E+. In view of Lemmas 4.1.8 and 4.1.9, we may now assume without loss of generality that T (t0) ≥ K > 0 for a compact operator K. Denote by S the closure of {T (t): t ≥ 0} in Lσ(E), the bounded linear operators on E endowed with the weak operator topology. Then, by Theorem 4.3.3, S contains a strictly positive projection Q commuting with every operator in S such that fix(T ) ⊆ F := QE and that QS := {QS : S ∈ S } ⊆ S is norm bounded group of positive operators with neutral element Q. Hence, the restrictions of the elements of S to F are a group of lattice isomorphisms. If x ∈ F , then |x| = |Qx| ≤ Q|x| and since ϕ is also a strictly positive fixed point of Q∗, it follows that hϕ, Q|x| − |x|i = 0 and hence Q|x| = |x|. Thus, F is a closed sublattice of E and hence a Banach lattice. Moreover, the norm on F is order continuous as every monotone order bounded sequence in F converges (see [49, Thm 2.4.2]). Since Q is strictly positive and the quasi-interior point e belongs to F , we have that QK|F e > 0. Therefore, T (t0)|F = QT (t0)|F dominates the non-trivial compact operator QK|F . If F were diffuse, compact operators would be disjoint from lattice isomorphisms by Theorem 4.1.10. Thus, there exists at least one atom a ∈ F . Since a ∈ fix(T ) = span{e} by Lemma 4.3.5, we obtain that e is an atom of F and consequently,

F = Fe = span{e} = ﬁx(T ).

It remains to check that z ∈ FC = QCEC. Then z ∈ ﬁx(TC) and therefore α = 0. ∗ Let w ∈ (EC) such that |hT (t)Cz, wi| = hz, wi = kzk > 0. Since QC is in the closure of TC in Lσ(EC), there exists t ≥ 0 such that

|hT (t)Cz − QCz, wi| < kzk.

65 4. Kernel and Harris operators

This implies that

|hQCz, wi| ≥ |hT (t)Cz, wi| − |hT (t)Cz − QCz, wi| > 0

∗ and hence QCz 6= 0. Now choose w ∈ (EC) vanishing on z and let ε > 0. Again, there exists t ≥ 0 such that

|hT (t)Cz − QCz, wi| < ε

which implies that

|hQCz, wi| ≤ |hT (t)Cz − QCz, wi| + |hT (t)Cz, wi| < ε.

Since ε > 0 was arbitrary, hQCz, wi = 0. Thus, by the Hahn-Banach theorem, QCz ∈ spanC{z} and therefore QCz = z ∈ FC as QC is a projection. We show next that Theorem 4.3.7 holds true for Abel bounded semigroups of Harris operators.

Proposition 4.3.8. Let T = (T (t))t≥0 be a positive, Abel bounded and irreducible strongly continuous semigroup with generator A. Suppose that T (t0) is a Harris operator for some t0 > 0. Then σp(AC) ∩ iR ⊆ {0}.

Proof. Assume that T (t)Cz = exp(iαt)z for some z ∈ EC\{0}, α ∈ R and all t ≥ 0. Since |z| ≤ T (t)|z| for all t ≥ 0, it follows from Lemma 4.3.1 that there exists a strictly positive linear form ϕ ∈ ﬁx(T ∗). Now, we endow E with the order continuous lattice norm kxkϕ := hϕ, |x|i and denote by (F, k · kF ) the completion ∗∗ of (E, k · kϕ), i.e. the closure of E in the bidual (E, k · kϕ) . Then T uniquely extends to a positive strongly continuous contraction semi- ˜ ˜ ˜ ˜ group T = (T (t))t≥0 on F . We show that T is irreducible. Let J be a closed ˜ ˜ and T -invariant ideal in (F, k · kF ). Then J := J ∩ E is a closed and T -invariant ideal in (E, k · k) and hence J = {0} or J = E. Since ϕ is order continuous, E is an ideal in F by [55, Lem IV 9.3]. This implies that J˜ is the closure of J in F and hence either J˜ = {0} or J˜ = F . Thus, T˜ is irreducible. Next, we verify that T˜(t0) is a Harris operator. By assumption, T := T (nt0) 6∈ (E∗ ⊗ E)⊥ for some n ∈ N. In view of Nakano’s carrier theorem [49, Thm 1.4.11], ∗ ∗ the strictly positive ϕ is a weak order unit of E = Eoc. It hence follows from Lemma 4.1.3 that (T ∧ (ϕ ⊗ e))x > 0 for some x ∈ E+. Since the extensions ϕ˜ ⊗ e and T˜ leave the ideal E invariant, we conclude that

(T˜ ∧ (ϕ ˜ ⊗ e))x = inf{T˜(x − y) +ϕ ˜(y)e : y ∈ F , 0 ≤ y ≤ x} F = inf{T (x − y) + ϕ(y)e : y ∈ E, 0 ≤ y ≤ x} E = (T ∧ (ϕ ⊗ e))x > 0.

66 4.3. Triviality of the peripheral point spectrum

This shows that T˜ 6∈ (F ∗ ⊗ F )⊥. Finally, we conclude from Theorem 4.3.7 that σp(AC)∩iR ⊆ σp(A˜C)∩iR ⊆ {0} where A˜ is the generator of T˜.

In what follows, we study the peripheral point spectrum of time-discrete semigroups. Let Γ := {z ∈ C : |z| = 1} denote the unit circle. In analogy with Theorem 4.3.7, it is natural to ask whether σp(TC) ∩ Γ ⊆ {1} for every power-bounded and irreducible kernel or compact operator T ∈ L (E)+. The following example shows that this conjecture is false even for matrices. Example 4.3.9. On E = R2, consider the matrix

0 1 T := (4.3.2) 1 0

Then T is a positive, irreducible, contractive and compact kernel operator but −1 ∈ σp(TC). However, by the arguments of Theorem 4.3.7 we obtain that the point spectrum of some power of TC is trivial. In the case where T is a kernel operator, this was ﬁrst proved in [34, Thm 3.6].

Theorem 4.3.10. Let T ∈ L (E)+ be an irreducible and power-bounded operator. If there exists m ∈ N such that T m ≥ K > 0 for a compact or a kernel operator n K ∈ L (E), then σp(TC ) ∩ Γ ⊆ {1} for some n ∈ N.

Proof. Let us assume that σp(TC) ∩ Γ 6= ∅, i.e. there exists α ∈ R and z ∈ EC such that z 6= 0 and TCz = exp(iα)z. Since

n n 0 < |z| = |exp(inα)z| = |TC z| ≤ T |z| (n ∈ N),

n it follows from Lemma 4.3.1, applied to the discrete semigroup T := (T )n∈N0 , that there exists a strictly positive ϕ ∈ fix(T ∗) and fix(T ) = span{e} where e := |z| is a quasi-interior point of E+. In view of Lemma 4.1.9, we may now assume that T m ≥ K > 0 for a compact operator K. As in the proof of Theorem 4.3.7, we conclude from Theorem 4.3.3 that there exists a strictly positive projection Q: E → E with the following properties: Q commutes with T and its range F := QE is a closed sublattice of E with order continuous norm that includes fix(T ). Moreover, the restriction of T to F is k Z m a lattice isomorphism with sup{kT|F k : k ∈ } < ∞. On the other hand, T|F dominates the non-trivial compact operator QK|F and it follows from Theorem 4.1.10 that F is not diffuse.

67 4. Kernel and Harris operators

We show that T|F is irreducible. Let J ⊆ F be a closed and T -invariant ideal with corresponding band projection P : F → J and let e∗ := P e. Since T e∗ ∈ J, we observe that

T e∗ = T P e = P T P e ≤ P T e = P e = e∗.

It follows from hϕ, e∗ − T e∗i = 0 that e∗ ∈ ﬁx(T ) = span{e} as ϕ is strictly positive. Hence, e∗ = 0 or e∗ = e which implies that either J = {0} or J = F . Now, we conclude from Lemma 4.3.6, applied to the restriction of T to F , that n n there exists n ∈ N such that F ⊆ ﬁx(T ). Let exp(iβ) ∈ σp(TC ) and denote by n ξ1, . . . , ξn the nth roots of exp(iβ), i.e. (ξk) = exp(iβ) for all 1 ≤ k ≤ n. Then we infer from

n exp(iβ) − TC = (ξ1 − TC)(ξ2 − TC) ... (ξn − TC)

that ξk ∈ σp(TC) for at least one 1 ≤ k ≤ n. Pick y ∈ EC\{0} such that TCy = ξky. Now, we obtain that y ∈ FC = QCEC by the same arguments as in the proof of k Theorem 4.3.7. Indeed, since QC is in the closure of TC = (TC )k∈N0 in Lσ(EC), ∗ for w ∈ (EC) satisfying hw, yi = kyk we ﬁnd some j ∈ N0 such that

j |hTCy − QCy, wi| < kyk.

j j As |hTCy, wi| = |ξkhy, wi| = kyk, it follows that |hQCy, wi| > 0 and hence QCy 6= 0. ∗ On the other hand, for every w ∈ (EC) vanishing on y and for every ε > 0 there j exists j ∈ N0 such that |hTCy − QCy, wi| < ε and hence |hQCy, wi| < ε. Thus, QCy ∈ span{y} which implies that QCy = y ∈ FC since QC is a projection. Altogether, we proved that

n n y = TC y = ξk y = exp(iβ)y

n and hence exp(iβ) = 1. Since exp(iβ) ∈ σp(TC ) was arbitrary, this completes the proof.

Theorem 4.3.10 holds true for irreducible and Abel bounded Harris operators T ∈ L (E)+. This can be proved analogously to Proposition 4.3.8.

Remark 4.3.11. (a) It is well-known that σp(TC) ∩ Γ ⊆ {1} for every power- bounded operator T ∈ L (E)+ that is expanding (or strongly positive) i.e. T x is a weak unit for every x ∈ E+ \{0}, c.f. [55, Prop V 5.6]. This can also be obtained from the proof of Theorem 4.3.10 by observing that such an operator T is irreducible and its restriction to F is again strongly positive and therefore no lattice isomorphism unless F is one-dimensional.

68 4.3. Triviality of the peripheral point spectrum

(b) The assumption that T be irreducible cannot be dropped in Theorem 4.3.7 (and Theorem 4.3.10). Indeed, let (T1(t))t≥0 be a positive and bounded strongly continuous semigroup on E such that T1(t0) ≥ K > 0 for some compact operator K and some t0 > 0. Let (T2(t))t≥0 be another positive and bounded strongly continuous semigroup on E such that iα is an eigenvalue of its generator for some α 6= 0. Then T (t) := T1(t) ⊕ T2(t) deﬁnes a positive and bounded strongly continuous semigroup on E ⊕ E where T (t0) dominates the compact operator K ⊕ 0 but iα is an eigenvalue of its generator.

(c) If A is the generator of a bounded and positive strongly continuous semigroup T = (T (t))t≥0 such that T (t0) is compact for some t0 > 0, it is well-known and easy to see that σ(AC) ∩ iR ⊆ {0}. Indeed, in this case T is eventually norm continuous [23, Lem II 4.22] and hence σ(AC) ∩ iR is bounded [23, Thm II 4.18]. Since the boundary spectrum is cyclic by [32, Thm 2.4], it follows that σ(AC) ∩ iR ⊆ {0}. In view of Remark (c) one may ask if even the peripheral spectrum is trivial under the assumptions of Theorem 4.3.7 and Theorem 4.3.10. We conclude this section with an example of a time-discrete semigroup showing that this can fail. Example 4.3.12. Let E := `p for a fixed 1 < p < ∞. We construct a positive and n irreducible contraction T ∈ L (E) such that σp(TC) = ∅ and σ(TC ) ∩ Γ = Γ for all n ∈ N. Since E is atomic, T is automatically a kernel operator by Lemma 4.1.5 and hence satisfies the assumptions of Theorem 4.3.10. Q∞ 1 Let (bn) ⊆ (0, 1) be a decreasing sequence with n=1(1 − bn) = 2 and let p P∞ 0 ≤ an ≤ 1 − (1 − bn) be small enough such that 0 < n=1 an ≤ 1. For p x = (xn) ∈ ` we define

∞ X T x := anxn, (1 − b1)x1, (1 − b2)x2, (1 − b3)x3,... . n=1

It follows from Jensen’s inequality that p ∞ p ∞ a ∞ p X X n X p p kT xk = ak P∞ · xn + (1 − bn) |xn| ak k=1 n=1 k=1 n=1 ∞ p ∞ ∞ X X an p X p p ≤ ak P∞ · |xn| + (1 − bn) |xn| ak k=1 n=1 k=1 n=1 ∞ X p p p ≤ (an + (1 − bn) )|xn| ≤ kxk , n=1

p for all x = (xn) ∈ ` . Thus, T is a contraction.

69 4. Kernel and Harris operators

It is well-known that the complexiﬁcation of E equals `p(C), the p-summable sequences in C. Now, assume that there is λ ∈ σp(TC) and denote by z = (zn) ∈ EC, z 6= 0, a corresponding eigenvector. Since TC is injective and contractive, 0 < |λ| ≤ 1. By applying the operator n + 1 times we observe that 1 z = (1 − b )(1 − b ) ... (1 − b )z n+1 λn 1 2 n 1 for all n ∈ N. Hence, z is not a null sequence, which is impossible. This shows that σp(TC) = ∅. In order to calculate the peripheral spectrum of TC, we ﬁrst point out that the n discrete semigroup (TC )n∈N0 is not strongly stable. Indeed, for e1 = (1, 0, 0,... ) we have that n Y 1 kT ne k ≥ (1 − b ) ≥ 1 k 2 k=1 for all n ∈ N. Thus, it follows from the characterization of strong stability by Arendt, Batty, Lyubich and V˜u[10, Thm 5.1] that σ(TC)∩Γ is uncountable. Since σ(TC) ∩ Γ is cyclic [55, Thm V 4.9], it is dense in Γ and hence σ(TC) ∩ Γ = Γ. n By factorizing λ − TC as in the proof of Theorem 4.3.10 we observe that also n σ(TC ) ∩ Γ = Γ for all n ∈ N.

70 CHAPTER 5 Stability of semigroups

Most results on the asymptotic behavior of semigroups establish a connection to spectral properties of their generators. As one of the most important results of that type let us mention the following theorem that was proven independently by Arendt and Batty [10] and Lyubich and V˜u[48], see also [8, 3.3.2]. If (T (t))t≥0 is a bounded and mean ergodic strongly continuous semigroup with generator ∗ A on a Banach space X such that σp(A ) ∩ iR ⊂ {0} and σ(A) ∩ iR is at most countable, then limt→∞ T (t)x exists for all x ∈ X. An analog result holds for time-discrete semigroups. Here, we study the asymptotic behavior of positive semigroups with a different technique developed by Greiner. He proved in [33, Kor 3.11] that a positive and contractive strongly continuous semigroup on Lp converges strongly to an equilibrium if it contains a kernel operator and admits a fixed point that is a quasi-interior point; see also [31] and [9, Thm 4.2]. In view of the above mentioned theorem, this is a remarkable result since the peripheral spectrum of a kernel operator can be uncountable as shown in Example 4.3.12. Under the additional assumption that the semigroup is irreducible, we generalize Greiner’s result in Section 5.1 to bounded semigroups of Harris operators on Banach lattices with order continuous norm. For time-discrete semigroups, the following analogous assertion holds true as proved in [34]. If T is a positive, power-bounded and irreducible Harris operator with non-trivial fixed space, then there exists some k ∈ N such that the subsequence T nk converges strongly as n tends to infinity. We show in addition that the sequence T n itself converges strongly if T is expanding, i.e. T x is a weak

71 5. Stability of semigroups

unit for all x > 0. We conclude this section with examples showing the optimality of the foregoing theorems. In the subsequent sections we apply the results of Section 5.1 to semigroups on spaces of measures. First, in Section 5.2, we give a purely analytic proof of Doob’s theorem on stability of certain semigroups operating on spaces of measures. In the version used here, this theorem is originally due to Stettner [58, Thm 1]; see also [56]. Our approach is similar to the one in [30]. However, due to the lattice structure of weakly continuous operators established in Sections 2.2 and 2.3, the proof now simpliﬁes as we can work directly within the lattice of weakly continuous operators. This proof of Doob’s theorem is going to be published in [28]. Finally, in Section 5.3, we present a Tauberian theorem for certain semigroups on spaces of measures, published in [27]. Under the assumption that the semigroup contains an operator with the strong Feller property, i.e. its adjoint maps bounded measurable functions to continuous ones, we prove that weak ergodicity – as characterized by Theorem 3.3.9 – already implies convergence of the semigroup in total variation norm. In comparison to Doob’s theorem from Section 5.2, this Tauberian theorem shows stability of semigroups which are not necessarily irreducible. In particular, their ﬁxed space can be of arbitrary high dimension.

5.1 Stability of semigroups of Harris operators

Throughout this section, let E be a Banach lattice with order continuous norm. In order to treat time-discrete and time-continuous semigroups simultaneously, we write (T (t))t∈R for both, where R is either N0 in the discrete case or [0, ∞) in the continuous case.

Deﬁnition 5.1.1. A positive semigroup (T (t))t∈R ⊂ L (E)+ – time-discrete or time-continuous – is said to have non-disjoint operators if there exist t, s ∈ R, n t 6= s, such that T (t) ∧ T (s) > 0. If a time-discrete semigroup S = {T : n ∈ N0} has non-disjoint operators, we also say that the operator T has non-disjoint powers.

Our main tool is Greiner’s zero-two law which we present as Theorem A.1 in the appendix. The following proposition is a consequence of it and a generalized version of [33, Kor 3.9].

Proposition 5.1.2. Let S = (T (t))t∈R ⊂ L (E)+ be a bounded and irreducible semigroup on E which is strongly continuous or time-discrete. Suppose that fix(T (t)) = fix(S ) 6= {0} for all t ∈ R and that S has non-disjoint operators. Then there exist a strictly positive x∗ ∈ fix(S ∗) and a quasi-interior point e ∈

72 5.1. Stability of semigroups of Harris operators

ﬁx(S ) of E+ such that lim T (t)x = hx∗, xie t→∞ for every x ∈ E.

Proof. By Lemma 4.3.1 there exists a strictly positive x∗ ∈ ﬁx(S ∗) and a quasi- ∗ interior point e of E+ such that ﬁx(S ) = span{e}. We may assume that hx , ei = 1. As S has non-disjoint operators,

E2 := {y ∈ E :(T (t) ∧ T (t + τ))|y| = 0 for all t ∈ R}= 6 E for some τ > 0. Since S is irreducible, it follows from Greiner’s zero-two law, Theorem A.1, that E2 = {0} and

lim |T (t) − T (t + τ)|e = 0. t→∞

Hence, |T (t)(I − T (τ))y| ≤ |T (t) − T (t + τ)|e → 0 (t → ∞) for all y ∈ [−e, e], i.e. lim T (t)z = 0 for all z ∈ D := (I −T (τ))[−e, e]. As D is total in (I − T (τ))E and S is bounded, limt→∞ T (t)x = 0 for all x ∈ (I − T (τ))E. By Lemma 4.3.2, T (τ) is mean ergodic and therefore, by Theorem 3.0.8, the mean ergodic theorem, E = ﬁx(T (τ)) ⊕ (I − T (τ))E. Since ﬁx(T (τ)) = span{e}, the corresponding mean ergodic projection is given by x∗ ⊗ e. This completes the proof.

The central assumption of Proposition 5.1.2 – S having non-disjoint operators – is rather technical and in general hard to verify. However, if the semigroup contains an irreducible Harris operator, this assumption holds automatically due to a theorem of Axmann presented as Theorem B.2 in the appendix. This gives rise to the following theorems, the main results of this section.

Theorem 5.1.3. Let T = (T (t))t≥0 ∈ L (E)+ be a bounded, irreducible and strongly continuous semigroup on E with generator A. Suppose that fix(T ) 6= {0} and that T (t0) is a Harris operator for some t0 > 0. Then there exist a strictly ∗ ∗ positive x ∈ fix(T ) and a quasi-interior point e ∈ fix(T ) of E+ such that

lim T (t)x = hx∗, xie t→∞ for all x ∈ E.

73 5. Stability of semigroups

Proof. By Lemma 4.3.1 there exists a strictly positive x∗ ∈ ﬁx(T ∗) and a quasi- interior point e of E+ such that ﬁx(T ) = span{e}. Theorem 4.3.7 implies that σp(AC) ∩ iR = {0}. Hence, for all t > 0

2πin ﬁx(T (t)) = span Z ker − A = ker A = ﬁx(T ) = span{e} n∈ t

by [23, Cor IV 3.8]. We prove next that T := T (t0) is irreducible. Let J ⊆ E be a T -invariant closed ideal. As the norm is order continuous, J is a projection band. Denote by P : E → J the corresponding band projection. Then T P e ∈ J and hence

T (I − P )e = e − T P e = e − P T P e ≥ e − P T e = (I − P )e.

Since ﬁxs(T ) = ﬁx(T ) by Lemma 4.1.8, it follows that

(I − P )e, P e ∈ ﬁx(T ) = span{e}.

Thus, J = {0} or J = E which shows that T is irreducible. Now it follows from Axmann’s theorem, Theorem B.2, applied to T that T has non-disjoint operators. An application of Proposition 5.1.2 completes the proof.

For an irreducible power-bounded Harris operator T ∈ L (E)+, we obtain strong convergence of a subsequence (T nk) for a ﬁxed n ∈ N. This is optimal in view of Example 4.3.9. If the operator is not only irreducible but even expanding, k the sequence (T )k∈N itself converges strongly to a projection of rank one.

Theorem 5.1.4. Let T ∈ L (E)+ be a power-bounded and irreducible Harris operator with fix(T ) 6= {0}. Then there exists n ∈ N such that T nk converges strongly as k tends to infinity. If T is even expanding, i.e. T x is a weak unit for all x > 0, then there exist ∗ ∗ a strictly positive x ∈ fix(T ) and a quasi-interior point e ∈ fix(T ) of E+ such k ∗ that limk→∞ T x = hx , xie. Proof. By Theorem B.2, T has non-disjoint powers, i.e. T a ∧ T b > 0 for some natural numbers a < b. For n := b − a we conclude as in the proof of Proposition 5.1.2 that E = fix(T n) ⊕ (I − T n)E

k n and limk→∞ T x = 0 for all x ∈ (I − T )E. If T is strongly positive, then σp(TC)∩Γ ⊆ {1} by Remark 4.3.11 (a). Therefore, ﬁx(T k) = ﬁx(T ) for all k ∈ N and the assertion follows immediately from Axmann’s theorem and Proposition 5.1.2.

74 5.1. Stability of semigroups of Harris operators

In view of Proposition 5.1.2 it would be interesting to ﬁnd more classes of positive operators with non-disjoint powers besides irreducible Harris operators treated in Axmann’s theorem. This would allow us to generalize Theorem 5.1.3 and 5.1.4 accordingly. One may ask, for instance, whether a positive operator T ∈ L (E)+ has non-disjoint powers under one of the following conditions.

(a) (A power of) T dominates a non-trivial compact operator and T is irreducible.

(b) (A power of) T is expanding.

In the following we provide examples showing that neither condition (a) nor (b) is sufficient for an operator to have non-disjoint powers. Example 5.1.5. We show that there exists a positive, compact and irreducible operator T on L2(Γ), the square-integrable functions on the unit circle endowed with the Lebesgue measure, such that T n ∧ T m = 0 for all n 6= m. Based on a work of Varopoulos [60], Arendt constructed a self-adjoint and compact Markov operator T on L2(Γ) such that T n ∧ T m = 0 whenever n 6= m [7, Ex 3.7]. By Theorem 5.1.6 below, there exists a T -invariant band B in L2(Γ) such that the restriction of T to B is irreducible (and still compact). Since B is of the form {f ∈ L2(Γ) : f = 0 on A} for some measurable A ⊆ Γ by [55,III §1 Ex.2] and Γ\A is not a nullset, B is in turn isomorphic to L2(Γ) by [15, Cor 6.6.7 and Thm 9.2.2]. Recall from Lemma 4.1.8 that fixs(T ) = fix(T ) if, for instance, fix(T ∗) contains a strictly positive functional. This is why the following theorem can be applied to the self-adjoint Markov operator of Example 5.1.5.

s Theorem 5.1.6. Let T ∈ L (E)+ be a compact operator such that fix (T ) = fix(T ). If there exists a quasi-interior point e ∈ fix(T ) of E+, then there are finitely many disjoint T -invariant bands B1,...,BN ⊆ E distinct from {0} such that E = B1 ⊕ · · · ⊕ BN and the restriction of T to Bk is irreducible for all k = 1,...,N.

Proof. Aiming for a contradiction, we assume that there is no T -invariant band A ⊆ E except {0} such that the restriction of T to A is irreducible. Then T is in particular not irreducible on E and hence there is a T -invariant closed ideal A1 distinct from {0} and E. As the norm on E is order continuous, every closed ideal is a projection band. Denote by P1 : E → A1 the corresponding band projection. It follows from TP1e = P1TP1e ≤ P1T e = P1e that T (I − P1)e ≥ (I − P1)e. Hence, by assumption, (I − P1)e ∈ ﬁx(T ) and by ⊥ linearity also P1e ∈ ﬁx(T ). Thus (I − P1)E = A1 is a non-trivial T -invariant

75 5. Stability of semigroups

band, too. We may assume that kek = 1 and 1 kP ek ≤ kek; 1 2 ⊥ otherwise we replace A1 with A1 and P1 with I − P1. By our assumption, T|A1 is not irreducible. Hence we ﬁnd a T -invariant band A2 ⊆ A1 distinct from {0} and A1 with band projection P2 : E → A2 such that 1 kP ek ≤ kP ek. 2 2 1

Inductively, we obtain a decreasing sequence An+1 ⊆ An of T -invariant bands with projections Pn : E → An satisfying

−n kPnek ≤ 2 kek

for all n ∈ N. Now consider the sequence xn := Pne/kPnek. Since T is compact, there exists a subsequence of (T xn) = (xn) converging to some x with kxk = 1. On the other hand, x ∈ An for every n ∈ N as xk ∈ An for all k ≥ n. Thus, ⊥ e = lim(I − Pn)e ∈ {x} which implies that x = 0 in contradiction to kxk = 1.

We proved the existence of a T -invariant band {0}= 6 B1 ⊆ E such that T|B1 is irreducible. If B1 6= E we may apply the same argument to the restriction of T ⊥ ⊥ to B1 to obtain a T -invariant band {0}= 6 B2 ⊆ B1 such that T|B2 is irreducible. Continue this construction inductively as long as B1 ⊕ · · · ⊕ Bn 6= E. Suppose that this process does not terminate after ﬁnitely many steps, i.e. we obtain an

infinite sequence Bn of disjoint non-trivial bands such that T|Bn is irreducible. Denote by Qn : E → Bn the corresponding band projections and let yn := Qne/kQnek. Then a subsequence of T yn = yn converges to some y ∈ E with kyk = 1 since T is compact. On the other hand, Qky = limn→∞ Qkyn = 0 for N ⊥ N every k ∈ implies that y ∈ Bk for all k ∈ . This shows that every yk is contained in {y}⊥ and so is y. Hence, y = 0 in contradiction to kyk = 1. We conclude that the process of constructing B1,B2,... ends after finitely many steps, which completes the proof. The following example is due to Jochen Glück. Example 5.1.7. Let E := L1([0, 1)), the integrable functions on the interval [0, 1) with respect to the Lebesgue measure λ. We construct an expanding operator ∗ n T ∈ L (E)+ with T 1 = 1 and T 1 = 1 such that (T ) does not convergence strongly. In particular, T is not a Harris operator by Theorem 5.1.4 and all of its powers are disjoint by Proposition 5.1.2. For n ∈ N and k = 1,..., 2n we consider the dyadic intervals k − 1 k I := , . n,k 2n 2n

76 5.1. Stability of semigroups of Harris operators

Then [0, 1) is the disjoint union ∪˙ k=1,...,2n In,k for every n ∈ N. Let S2n be the n symmetric group on 2 elements. Every permutation σ ∈ S2n induces a measure preserving transformation σ˜ : [0, 1) → [0, 1) that simply maps each interval In,k to In,σ(k). Now we define Tσ ∈ L (E) by Tσf := f ◦ σ˜ for every σ ∈ S2n and Tn ∈ L (E) by 1 X T := T n 2n! σ σ∈S2n for every n ∈ N. Next, we define an operator T˜ as a weighted sum of the operators Tn in the following way: For a strictly decreasing sequence (αk) ⊂ (0, ∞) such P∞ 1 N that k=1 αk = 4 we set βk := αk − αk+1 for k ∈ and β0 := 1 − α1. Then P∞ P∞ k=0 βk = 1 and k=n βk = αn. Now we define

∞ X T˜ := βnTn. n=1

It is easy to check that every Tσ, every Tn, and hence also T˜, is a positive operator on E such that T˜1 = 1 and T˜∗1 = 1. Now we show that T˜ is expanding. ˜ Let A, B ∈ B([0, 1)) with λ(A), λ(B) > 0. We prove that (T 1A)1B > 0. In a first step we show that there exists n0 ∈ N such that for all n ≥ n0 we find n k ∈ Kn := {1,..., 2 } satisfying 3 3 λ(A ∩ I ) ≥ · λ(I ) = · 2−n. (5.1.1) n,k 4 n,k 4 Aiming for a contradiction, we assume the contrary and denote by M ⊂ N the set of those numbers n ∈ N such that (5.1.1) fails to holds for every k ∈ Kn. Let 1 0 < ε < 4 λ(A) and choose, by the regularity of λ, an open set U ⊃ A such that λ(U \ A) < ε. Since M is assumed to be infinite, we find

J ⊂ {(n, k) ∈ M × N : k ∈ Kn} such that U is the disjoint union [˙ U = In,k. (n,k)∈J Then we obtain the contradiction X 3 1 λ(U) < λ(A ∩ U) + ε = λ(A ∩ I ) + ε < · λ(U) + · λ(A) ≤ λ(U). n,k 4 4 (n,k)∈J

We proved that for eventually all n ∈ N assertion (5.1.1) holds for some k ∈ Kn. Applying the same arguments to the set B, we conclude that there exists n ∈ N

77 5. Stability of semigroups

and k1, k2 ∈ Kn such that 3 3 λ(A ∩ I ) ≥ · 2−n and λ(B ∩ I ) ≥ · 2−n. n,k1 4 n,k2 4

Pick σ ∈ S2n such that σ(k2) = k1. Then Z Z T (1 )1 dλ ≥ T (1 ) dλ σ A B σ A∩In,k1 [0,1) B∩In,k2 Z = 1 dλ A∩In,k1 σ˜(B∩In,k2 ) 1 = λ(A ∩ I ∩ σ˜(B ∩ I )) ≥ · 2−n > 0. n,k1 n,k2 2

n −1 Since T˜ ≥ βn · (2 !) · Tσ, we conclude that Z T˜(1A)1B dλ > 0. [0,1)

This shows that T˜ is expanding. Next, we modify T˜ such that it becomes not strongly stable. To this end, we consider the measure preserving transformation ϕ: [0, 1) → [0, 1) given by ϕ(ω) := 2ω mod 1 and the induced Koopman operator Tϕf := f ◦ ϕ. Now we deﬁne the desired operator T := Tϕ ◦ T˜. Obviously, T is positive, satisfying T 1 = 1 and T ∗1 = 1. Since

−1 µ({Tϕf = 0}) = µ(ϕ ({f = 0})) = µ({f = 0})

for every f ∈ E, Tϕ maps weak units to weak units. Thus, T is expanding. n Now we show that the sequence (T 1[0,1/2)) does not converge. For n ∈ N we deﬁne the sets [ k k + 1 A := , . n 2n 2n k∈{0,...,2n−1} k even

1 1 1 −1 1 N Then Tϕ An = An ◦ ϕ = ϕ (An) = An+1 for all n ∈ . Moreover, for all 1 1 1 1 n > k ≥ 0 and every σ ∈ S2k we have Tσ An = An and therefore Tk An = An . Now we prove by induction that

n n1 X 1 T A1 ≥ 1 − αk An+1 (5.1.2) k=1

78 5.2. Doob’s theorem

for all n ∈ N0. Assume that (5.1.2) holds for some n ∈ N0. Then n n n n+11 X 1 X X 1 T A1 ≥ T 1 − αk An+1 ≥ 1 − αk Tϕ βkTk An+1 k=1 k=1 k=0 n n n X X 1 X 1 = 1 − αk βk · An+2 = 1 − αk (1 − αn+1) An+2 k=1 k=0 k=1 n+1 X 1 ≥ 1 − αk An+2 . k=1 n1 n1 3 1 N 1 We showed that T [0,1/2) = T A1 ≥ 4 An+1 for all n ∈ . Since µ(An+1) = 2 n for all n ∈ N and the only possible limit of the sequence (T 1[0,1/2)) would be 1 1 2 · , we conclude that the sequence does not converge.

5.2 Doob’s theorem

In the following we give a proof of Doob’s theorem on stability of semigroups of weakly continuous operators on the space of all measures based on the lattice structure established in Sections 2.2 and 2.3. This is part of [28], a joint work with Markus Kunze. Let Ω be a Polish space endowed with its Borel σ-algebra B(Ω). We denote by M (Ω) the space of all signed measures on B(Ω). Recall that a weakly continuous operator T ∈ L (M (Ω), σb) with associated transition kernel k is called Markovian if k(x, · ) is a probability measure for all x ∈ Ω. A semigroup S ⊂ L (M (Ω), σb) is said to Markovian if it consists of Markovian operators. We adopt the following terminology from [18].

Deﬁnition 5.2.1. Let T = (T (t))t≥0 ⊂ L (M (Ω), σb) be a time-continuous semigroup of weakly continuous operators. If lim(T (t)f)(x) = f(x) t→0 for all f ∈ Cb(Ω) and all x ∈ Ω, then T is said to be stochastically continuous.

For t0 > 0, the semigroup T is called t0-regular if the measures kt0 (x, · ) and kt0 (y, · ) are equivalent for all x, y ∈ Ω.

Remark 5.2.2. If T = (T (t))t≥0 is t0-regular for some t0 > 0, then T is s-regular for all s ≥ t0 and the measures ks(x, · ) and kt(y, · ) are equivalent for all s, t ≥ t0 and x, y ∈ Ω. Indeed, for A ∈ B(Ω) and r > 0 we have Z

kt0+r(x, A) = (T (t0 + r)δx)(A) = kt0 (y, A)kr(x, dy). (5.2.1) Ω

79 5. Stability of semigroups

Thus, kt0+r(x, · ) kt0 (y, · ) for all r > 0, x ∈ Ω and y ∈ Ω. Conversely, if

(T (t0 + r)δx)(A) = kt0+r(x, A) = 0

for some A ∈ B(Ω), it follows from (5.2.1) that kt0 (y, A) = 0 for some and hence all y ∈ Ω. Now we are able to prove the announced stability result, which is originally due to Stettner [58, Thm 1] who gave a probabilistic proof.

Theorem 5.2.3 (Doob). Let T = (T (t))t≥0 ⊂ L (M (Ω), σb) be a stochastically continuous Markovian semigroup and let µ ∈ M (Ω) be an invariant probability measure. If T is t0-regular for some t0 > 0, then

lim T (t)ν = ν(Ω) · µ t→∞

in the norm of M (Ω). Moreover, µ is the unique invariant probability measure and equivalent to all kt(x, · ) for t ≥ t0 and x ∈ Ω.

Proof. Let us denote by E := {µ}⊥⊥ the band generated by the invariant measure µ, which consists precisely of those measures that are absolutely continuous with respect to µ, see [3, Thm 10.61]. In view of the equality Z µ = T (t)µ = kt(x, · ) dµ(x) Ω

it follows from the t0-regularity of T and Remark 5.2.2 that µ is equivalent to kt(x, · ) for all x ∈ Ω and t ≥ t0. Hence, for every measure ν > 0, Z

T (t0)ν = kt0 (x, · ) dν(x) Ω

is equivalent to µ, so that T (t0)ν ∈ E for every measure ν > 0. Replacing ν by T (t0)ν, it therefore suﬃces to show that

lim T (t)ν = ν(Ω) · µ t→∞ for all ν ∈ E. Let S(t) := T (t)|E denote the restriction of T (t) to E. Then the semigroup S := (S(t))t≥0 is clearly contractive and positive. Moreover, it is strongly continuous by [36, Thm 4.6]. Fix r > s ≥ t0. Since the measures ks(x, · ) and kr(x, · ) are equivalent for all x ∈ Ω, they cannot by disjoint. Hence, the operator

S := S(s) ∧ S(r) = (T (s) ∧ T (r))|E

80 5.3. A Tauberian theorem for strong Feller semigroups is not zero as, by Theorem 2.2.6, it is weakly continuous and given by the transition kernel q(x, · ) := ks(x, · ) ∧ kr(x, · ) satisfying q(x, Ω) > 0 for all x ∈ Ω. Thus, the semigroup S has non-disjoint operators. In order to prove that the fixed space fix(S(t)) is independent of t > 0, we first show that for t ≥ t0 the operator S(t) is expanding. Indeed, since two measures µ1, µ2 ∈ M (Ω) are disjoint if and only if |µ1|(B) = |µ2|(Ω \ B) = 0 for some B ∈ B(Ω) by [3, Lem 10.58], for every t ≥ t0 and every ν ∈ E+ \{0} the measure S(t)ν, which is equivalent to µ, is a weak unit of E. Since the semigroup S consists of expanding operators, it follows from Remark [26, Rem 3.5(b)] that σp(G) ∩ iR ⊂ {0} where G denotes the generator of S . Thus, by [23, IV 3.8] one has that

2πin ﬁx(S(t)) = span ker − G = ker G n∈Z t for all t > 0. Since S is irreducible as it consists of expanding operators, we proved that S satisﬁes the assumptions of Proposition 5.1.2 and the assertion follows.

Applying Proposition 5.1.2 to a discrete semigroup yields the following result for Markov chains.

Theorem 5.2.4. Let T ∈ L (M (Ω), σb) be a Markovian operator with associated transition kernel k such that the measures k(x, · ) and k(y, · ) are equivalent for n x, y ∈ Ω. If there exists a T -invariant probability measure µ, then limn→∞ T ν = ν(Ω) · µ for every ν ∈ M (Ω) in total variation norm.

Proof. For E := {µ}⊥⊥, one concludes as in the proof of Theorem 5.2.3 (and Remark 5.2.2) that T M (Ω) ⊂ E and that S := T|E is expanding with non-disjoint n powers. Thus, σp(S)∩Γ ⊂ {1} by Remark 4.3.11 (a) and therefore ﬁx(S) = ﬁx(S ) for all n ∈ N. Now the assertion follows from Proposition 5.1.2.

5.3 A Tauberian theorem for strong Feller semigroups

In this section we prove that for eventually strong Feller semigroups on the space of all Borel measures weak ergodicity is already suﬃcient for stability, i.e. pointwise convergence of the semigroup in total variation norm. All results of this section have been published in [27]. Our strategy is to use Bukhvalov’s theorem to show that the square of every strong Feller operator, which is ultra Feller by [52, §1.5], is a kernel operator.

81 5. Stability of semigroups

Hence, by restricting the semigroup to a smaller space we obtain an irreducible strongly continuous semigroup to which Theorem 5.1.3 can be applied. Throughout, Ω denotes a Polish space which we endow with its Borel σ-algebra B(Ω).

Theorem 5.3.1. Let T ∈ L (M (Ω), σb) be an ultra Feller operator. Then T is a kernel operator.

⊥⊥ ⊥⊥ Proof. Let µ ∈ M (Ω)+ and ν := T µ, then T {µ} ⊂ {ν} . Let us denote ⊥⊥ ⊥⊥ by Tµ the restriction of T to {µ} . By the Radon-Nikodym theorem, {µ} and {ν}⊥⊥ are isometrically isomorphic to L1(Ω, µ) and L1(Ω, ν). Thus, we 1 1 may consider Tµ as an operator from L (Ω, µ) to L (Ω, ν) and we prove that ∗ ∞ ∞ Tµ : L (Ω, ν) → L (Ω, µ) is a kernel operator by applying Bukhvalov’s theorem in the version of Theorem 4.2.12. It is easy to check that

∗ 0 Tµ [f] = [T f]

∞ for every f ∈ Bb(Ω), where [f] denotes the equivalence class of f in L (Ω, µ) and 0 T the σb-adjoint of T . ∞ Let (fn) ⊂ L (Ω, ν) be a bounded sequence such that limkfnkL1(Ω,ν) = 0. By choosing representatives we may assume that every fn is a bounded measurable function. Moreover, we may assume that each fn vanishes on Ω \ supp(ν). Then

∗ 0 0 = limhfn, νi = limhTµ fn, µi = limhTµfn, µi.

0 Let ω ∈ Ω such that (Tµfn)(ω) does not converge to 0. Then there exists ε > 0 0 N and a subsequence (fnk ) of (fn) such that Tµfnk (ω) ≥ ε for all k ∈ . By the ultra Feller property of Tµ, the family 0 N {Tµfnk : k ∈ } is equicontinuous. Therefore, we ﬁnd an open neighborhood U of ω such that 0 N (Tµfnk )(s) ≥ ε/2 for all s ∈ U and k ∈ . Now we conclude from Z ε 0 µ(U) ≤ Tµfnk dµ → 0 (k → ∞) 2 Ω 0 that µ(U) = 0 and hence U ⊂ Ω \ supp(µ). This proves that (Tµfn)(ω) converges to 0 for all ω ∈ supp(µ) and hence almost everywhere. Thus, it follows from Theorem 4.2.12 that

∗ ∞ ∗ ∞ ⊥⊥ Tµ ∈ (L (Ω, ν)oc ⊗ L (Ω, µ)) . By [49, Prop 1.4.15], the order continuous functionals on L∞(Ω, ν) are precisely 1 ∗ 1 ∞ ⊥⊥ L (Ω, ν). Thus, Tµ ∈ (L (Ω, ν) ⊗ L (Ω, µ)) .

82 5.3. A Tauberian theorem for strong Feller semigroups

∞ 1 ⊥⊥ ∗ Now we prove that Tµ ∈ (L (Ω, µ) ⊗ L (Ω, ν)) . Let 0 ≤ Sα ≤ Tµ , α ∈ Λ, 1 ∞ be an upwards directed net and Rα ∈ L (Ω, ν) ⊗ L (Ω, µ) such that sup Sα = Tµ ∗ ∗ ∞ 1 and Sα ≤ Rα for all α ∈ Λ. Then Sα ≤ Rα ∈ L (Ω, µ) ⊗ L (Ω, ν) for all α ∈ Λ. Since L1(Ω, ν) is an ideal in the dual of L∞(Ω, ν), we obtain that

∗ 1 1 Sα |L1(Ω,µ) : L (Ω, µ) → L (Ω, ν).

Now it follows from

∗ ∗ suph(Tµ − Sα)f, gi = suphf, (Tµ − Sα)gi = 0

1 ∞ ∗ for all f ∈ L (Ω, µ) and g ∈ L (Ω, ν) that Tµ = sup Sα and therefore

∞ 1 ⊥⊥ T |L1(Ω,µ)= Tµ ∈ (L (Ω, µ) ⊗ L (Ω, ν)) .

∗ ⊥⊥ It follows that TPµ ∈ (M (Ω) ⊗ M (Ω)) for every µ ∈ M (Ω)+ where Pµ denotes the band projection onto {µ}⊥⊥. Thus,

T = sup{TPµ : µ ∈ M (Ω)+} belongs to (M (Ω)∗ ⊗ M (Ω))⊥⊥ which completes the proof.

Now let T = (T (t))t≥0 ⊂ L (M (Ω), σb) be a stochastically continuous Marko- vian semigroup. It follows from [47, Thm 6.2] that T is integrable on the norming dual pair (M (Ω),Bb(Ω)) in the sense of [47, Def 5.1], see also Example 3.1.6. In particular, by [47, Thm 5.8], for every t > 0 there exists a Markovian operator At ∈ L (M (Ω), σb) satisfying

1 Z t hAtµ, fi = hT (s)µ, fi ds t 0 for all µ ∈ M (Ω) and f ∈ Bb(Ω). In the following, we call the semigroup T Bb- ergodic if the average scheme (T , (At)t>0) is weakly ergodic on (M (Ω),Bb(Ω)). The following proposition ensures that for every initial distribution the part on the disjoint complement of ﬁx(T ) converges to zero if T is Bb-ergodic and eventually strong Feller.

Proposition 5.3.2. Let P denote the band projection onto ﬁx(T )⊥. If T is Bb-ergodic and T (t0) is strong Feller for some t0 > 0, then

lim PT (t)µ = 0 t→∞ for all µ ∈ M (Ω).

83 5. Stability of semigroups

Proof. First note that, since ﬁx(T )⊥⊥ is T -invariant, R(t) := PT (t) deﬁnes a semigroup. Obviously, every operator R(t) is positive and contractive. Fix µ ∈ M (Ω)+ and let α := lim kPT (t)µk = inf kR(t)µk. t→∞ t≥0

Aiming for a contradiction, we assume that α > 0. We pick t1 > 0 such that α ν := PT (t1)µ satisﬁes kνk < α + 2 . Since T is Bb-ergodic, lim Atν =: ν˜ exists with respect to the σ(M (Ω),Bb(Ω))-topology and ν˜ ∈ ﬁx(T ) by [29, Lem 4.5]. In particular, 1 Z t hν,˜ 1i = lim hT (s)ν, 1i ds = kνk ≥ α. t→∞ t 0

Let t ≥ t0. As PT (2t)ν and ν˜ are disjoint, there exists a Borel set B ⊂ Ω such that (PT (2t)ν)(B) =ν ˜(Ω \ B) = 0. α Since kT (2t)νk ≤ kνk < α + 2 and

kPT (2t)νk = kR(t1 + 2t)µk ≥ α, α it follows from the additivity of the total variation norm that k(I −P )T (2t)νk < 2 . α 01 01 Hence, (T (2t)ν)(B) < 2 . Let f := T (t) B and g := T (t) Ω\B. Since T (t) = T (t − t0)T (t0) is strong Feller and Markovian, f, g ∈ Cb(Ω)+ and f + g = 1. It follows from hν,˜ gi = 0 that A := suppν ˜ ⊂ {g = 0} = {f = 1},

i.e. 1A ≤ f. Thus, α hT (t)ν, 1 i ≤ hT (t)ν, fi = hT (2t)ν, 1 i < . A B 2 1 α Since t ≥ t0 was arbitrary, we conclude that hT (t)ν, Ai < 2 for all t ≥ t0 and hence 1 Z t α α = hν,˜ 1Ai = lim hT (s)ν, 1Ai ≤ . t→∞ t 0 2 This contradicts our assumption that α > 0. Thus, α = 0.

Remark 5.3.3. The assumption that T is eventually strong Feller cannot be dropped in Proposition 5.3.2. A counterexample is given by the rotation group on the Borel measures on the unit circle. Note that the situation is diﬀerent for time-discrete semigroups. If T is a positive and mean ergodic contraction on an L-space E and P is the band projection onto ﬁx(T )⊥, then it is possible to prove that lim kPT nxk = 0 n→∞

84 5.3. A Tauberian theorem for strong Feller semigroups for all x ∈ E. Combining Theorem 5.1.3 and the irreducible decomposition of Theorem 4.2.13, we obtain stability of T on the band spanned by its ﬁxed space.

Proposition 5.3.4. Suppose that T (t0) is a kernel operator for some t0 > 0. ⊥⊥ Then limt→∞ T (t)µ exists for all µ ∈ fix(T ) . Proof. Since every T (t) is a contraction and the total variation norm is strictly monotone on the positive cone M (Ω)+, for all µ ∈ fix(T ) it follows from |µ| = |T (t)µ| ≤ T (t)|µ| that T (t)|µ| = |µ|. Hence, fix(T ) is a sublattice. ⊥⊥ ⊥ Now let µ ∈ fix(T )+ and denote by P the band projection onto fix(T ) . Let D be a maximal disjoint system in fix(T )+. Since the total variation norm on M (Ω) is an L-norm, i.e. it is additive on the positive cone M (Ω)+, there exists an at most countable subset C ⊂ D such that µ ∈ C ⊥⊥. In fact, for ζ ∈ D ⊥⊥ let Pζ denote the band projection onto {ζ} . Then for every m ∈ N there exist 1 only finitely many ζ ∈ D such that kPζ µk ≥ m . This implies that there are at most countably many ζ ∈ D such that Pζ µ > 0. Let C := (ζk)k∈N := {ζ ∈ D : N Pζ µ > 0} and (µk)k∈N := (Pζk µ)k∈N for some N ⊂ . We omit the easier case where N is finite and continue the proof for the case where N = N. Since µ is a fixed point of T , the band {µ}⊥⊥ is T -invariant. By Theorem 4.2.13, we may assume that the restriction of T to {µ}⊥⊥ is irreducible. Moreover, since T is stochastically continuous, this restriction is strongly continuous by [36, ⊥⊥ Thm 4.6]. Thus, for each k ∈ N, the limit νk := limt→∞ T (t)µk ∈ {ζk} exists ⊥⊥ by Theorem 5.1.3 applied to the Banach lattice {ζk} . Pn Next, we show that τn := k=1 νk is a Cauchy sequence. For a given ε > 0 N P∞ choose n ∈ such that k=n+1kµkk < ε. Then m ∞ X X kτn − τmk = kνkk ≤ kµkk < ε k=n+1 k=n+1

⊥⊥ for all m > n. Therefore, τ := lim τm ∈ ﬁx(T ) exists. We prove that N P∞ lim T (t)µ = τ. Let ε > 0 and choose n ∈ such that k=n+1kµkk < ε. Since T (t)µk converges to νk we ﬁnd s > 0 such that kT (t)µk − νkk < ε/n for all t ≥ s and all 1 ≤ k ≤ n. Finally, we obtain that ∞ X kT (t)µ − τk ≤ kT (t)µk − νkk k=1 n ∞ X X ε ≤ kT (t)µ − ν k + 2kµ k < n · + 2ε k k k n k=1 k=n+1

85 5. Stability of semigroups

for all t ≥ s. This shows that limt→∞ T (t)µ = τ. Let us remark that Proposition 5.3.4 remains true for every positive and contractive semigroup T = (T (t))t≥0 on a Banach lattice with strictly monotone and order continuous norm such that the restriction of T to {x}⊥⊥ is strongly continuous for every x ∈ ﬁx(T )⊥⊥. Now we prove our main result.

Theorem 5.3.5. If T is Bb-ergodic and T (t0) is strong Feller for some t0 > 0, then limt→∞ T (t)µ exists for all µ ∈ M (Ω).

⊥ Proof. Let µ ∈ M (Ω)+ and denote by P the band projection onto ﬁx(T ) . By Proposition 5.3.2, there exists an increasing sequence tn > 0 such that 1 N kPT (tn)µk < n for all n ∈ . Deﬁne

⊥⊥ µn := (I − P )T (tn)µ ∈ ﬁx(T ) .

It follows from [52, §1.5] that T (2t0) is ultra Feller and therefore a kernel operator by Theorem 5.3.1. Thus, by Proposition 5.3.4, νn := limt→∞ T (t)µn exists in ⊥⊥ ﬁx(T ) for every n ∈ N. Hence, there exists an increasing sequence sn > 0 1 N such that kT (t)µn − νnk < n for all n ∈ and t ≥ sn. This implies that for every n ∈ N

kT (t + tn)µ − νnk ≤ kT (t)(I − P )T (tn)µ − νnk + kT (t)PT (tn)µk 2 ≤ kT (t)µ − ν k + kPT (t )µk < n n n n

for all t ≥ sn. Since 2 2 kν − ν k ≤ kν − T (s + t )µk + kT (s + t )µ − ν k < + n m n m m m m m n m

⊥⊥ for all m ≥ n,(νn) is a Cauchy sequence. Let ν := lim νn ∈ ﬁx(T ) . Then for every ε > 0 there exists n ∈ N such that

kT (t)µ − νk ≤ kT (t)µ − νnk + kνn − νk < ε

for all t ≥ tn + sn which proves the claim. Making use of the characterization of weak ergodicity in Theorem 3.3.9, we obtain the following Corollary.

Corollary 5.3.6. If T (t0) is strong Feller for some t0 > 0, then the following assertions are equivalent.

0 0 (i) ﬁx(T ) separates ﬁx(T ) := {f ∈ Cb(Ω) : T (t) f = f for all t ≥ 0}.

86 5.3. A Tauberian theorem for strong Feller semigroups

(ii) The semigroup T is weakly ergodic in the sense that limt→∞ Atµ exists in the σ(M (Ω),Cb(Ω))-topology for all µ ∈ M (Ω).

(iii) The semigroup T is Bb-ergodic.

0 (iv) limt→∞ T (t)µ exists for every µ ∈ M (Ω) and β0- limt→∞ T (t) f exists for every f ∈ Bb(Ω).

Proof. Let us assume (i) and pick µ ∈ M (Ω). It follows from Theorem 3.3.9 that there existsµ ˜ ∈ ﬁx(T ) such that

limhAtµ − µ,˜ fi = 0 for all f ∈ Cb(Ω), i.e. assertion (ii) holds. As explained in Example 3.1.6, one has that

lim k(T (t0) − I)Atµk = 0. t→∞

Since T (t0) is strong Feller, assertion (ii) implies that

0 lim hAtµ − µ,˜ fi = lim hAtµ − T (t0)Atµ, fi + hAtµ − µ,˜ T (t0) fi = 0 t→∞ t→∞ for all f ∈ Bb(Ω), i.e. T is Bb-ergodic. Next, let us assume (iii). By Theorem 5.3.5 lim T (t)µ exists for all µ ∈ M (Ω). Fix f ∈ Bb(Ω). Since T (2t0) is ultra Feller by [52, §1.5], the family

0 {T (t) f : t ≥ 2t0} is equicontinuous and hence, by the Arzel`a-Ascolitheorem [41, Thm 3.6], relatively 0 0 β0-compact. Thus, there exists a β0-convergent subnet (T (tα)f) of (T (t)f)t≥2t0 . 0 Let P be the ergodic projection of T and g := β0- lim T (tα) f ∈ Cb(Ω). Since

0 0 hg, µi = limhT (tα) f, µi = limhf, T (tα)µi = hf, P µi = hP f, µi for every µ ∈ M (Ω), it follows that g = P 0f. As we can apply this argument to 0 0 0 every subnet of (T (t)f)t≥2t0 , this shows that β0- lim T (t) f = P f. In order to prove that (i) follows from (iv), we assume that lim T (t)µ exists for each µ ∈ M (Ω). For f ∈ ﬁx(T 0) choose µ ∈ M (Ω) such that hµ, fi =: α 6= 0. Letµ ˜ := lim T (t)µ ∈ ﬁx(T ). Then

hµ,˜ fi = lim hT (t)µ, fi = α 6= 0 t→∞ which shows that ﬁx(T ) separates ﬁx(T 0).

87 5. Stability of semigroups

Remark 5.3.7. This Tauberian theorem can also be applied to weakly ergodic strong Feller semigroups that are merely sub-Markovian, i.e. each kt(x, · ) is a positive measure with kt(x, Ω) ≤ 1. In this case, one extend Ω by an isolated point ∞ and define the extended semigroup T˜ by the transition kernels ˜ kt(x, · ) := kt(x, · ) + (1 − kt(x, Ω))δ∞ ˜ ˜ for x ∈ Ω and kt(∞, · ) := δ∞. Then T is still strong Feller and the fixed spaces ˜ ˜0 0 fix(T ) = fix(T ) ⊕ span{δ∞} and fix(T ) = fix(T ) ⊕ span{1∞} still separate each other. Now, the Corollary 5.3.6 states that T˜ and hence also T is stable.

88 Appendix

The main tools for our study of the asymptotic behavior of semigroups on Banach lattices are a zero-two law by Greiner [33, Thm 3.7] and a theorem by Axmann [14, Satz 3.5]. In order to be more self-contained and to improve the accessibility of both results, we present their proofs in what follows.

A Greiner’s zero-two law

We present the proof of Greiner’s zero-two law from [33] in a reformulation for Banach lattices with order continuous norm and without any continuity condition on the semigroup. Throughout, let E be a Banach lattice with order continuous norm. Using the notation from Section 5.1, we denote by T = (T (t))t∈R a positive and bounded semigroup on E which is either time-discrete or time-continuous. Fix some τ ∈ R such that τ > 0.

Theorem A.1 (Greiner’s zero-two law). Assume that ﬁx(T ) contains a quasi- ∗ interior point e of E+ and that there exists a strictly positive element in ﬁx(T ). Then

E2 := {y ∈ E :(T (t) ∧ T (t + τ))|y| = 0 for all t ∈ R}

⊥ and E0 := E2 are T -invariant bands. Moreover, if P denotes the band projection onto E2, then |T (t) − T (t + τ)|P e = 2P e for all t ∈ R and lim |T (t) − T (t + τ)|(I − P )e = 0. t→∞ To simplify notation, for t ∈ R we deﬁne the positive operators

S(t) := T (t) ∧ T (t + τ) and D(t) := |T (t) − T (t + τ)|

89 Appendix

on E. It follows immediately from this deﬁnitions that

1 S(t)x + D(t)x = x (A.1) 2

for all x ∈ ﬁx(T ) and t ∈ R. Further properties of S(t) and D(t) are provided by the following lemma.

Lemma A.2. Suppose that there exists a strictly positive functional in ﬁx(T ∗). Then the following assertions hold.

(a) D(t)T (s) ≥ D(t + s) and T (s)D(t) ≥ D(t + s) for all t, s ∈ R. Moreover, limt→∞ D(t)x ∈ ﬁx(T ) for all x ∈ ﬁx(T ).

(b) S(t)T (s) ≤ S(t + s) and T (s)S(t) ≤ S(t + s) for all t, s ∈ R. Moreover, limt→∞ S(t)x ∈ ﬁx(T ) for all x ∈ ﬁx(T ).

m (c) If limt→∞ S(t)x > 0 for all x ∈ ﬁx(T ) \{0}, then limt→∞ S(t) x > 0 for all m ∈ N and all x ∈ ﬁx(T ) \{0}.

Proof. (a) For all t, s ∈ R we have that

D(t)T (s) = |T (t) − T (t + τ)| · |T (s)| ≥ |(T (t) − T (t + τ))T (s)| = D(t + s)

and similarly that T (s)D(t) ≥ D(t + s). Let x ∈ ﬁx(T )+ and s ∈ R. Then

D(t)x = D(t)T (s)x ≥ D(t + s)x

for all t ∈ R. Hence, by the order continuity of the norm, y := lim D(t)x exists in E and T (s)y = lim T (s)D(t)x ≥ lim D(t + s)x = y ≥ 0. t→∞ t→∞ Since fixs(T (s)) = fix(T (s)) by Lemma 4.1.8, we conclude that y ∈ fix(T (s)). As s ∈ R was arbitrary, y ∈ fix(T ) as desired. (b) For all t, s ∈ R we have that

1 S(t)T (s) = (T (t + s) + T (t + τ + s) − D(t)T (s)) 2 1 ≤ (T (t + s) + T (t + τ + s) − D(t + s)) = S(t + s) 2 and similarly that T (s)S(t) ≤ S(t + s). Hence, S(t)x is increasing for all positive 1 x ∈ fix(T ) as t tends to infinity. Since 0 ≤ S(t) ≤ 2 (T (t) + T (t + τ)), we conclude as in the proof of part (a) that lim S(t)x exists in fix(T ) for all x ∈ fix(T ).

90 A. Greiner’s zero-two law

(c) Let x ∈ fix(T ) \{0} and define recursively xk := limt→∞ S(t)xk−1 for all k ∈ N where x0 := x. Then xk ∈ fix(T ) by part (b) and xk > 0 by assumption. It follows by induction that m m X m−j S(t) x − xm = S(t) (S(t)xj−1 − xj) j=1 N for all m ∈ and t ∈ R. As kS(t)k ≤ supt∈RkT (t)k, we conclude that m lim S(t) x = xm > 0 t→∞ for all m ∈ N. The key to the proof of the zero-two law is the following combinatorial lemma. Lemma A.3. For every m ∈ N m X m m 2 2−m − + 2 ≤ √ . j j − 1 m j=1 Proof. For k ∈ N and m = 2k − 1 we have m k−1 X m m Xm m − + 2 = 2 − + 2 j j − 1 j j − 1 j=1 j=1 m 2k = 2 = k − 1 k k Xm + 1 m + 1 = − + 1 j j − 1 j=1 m+1 1 X m + 1 m + 1 = − + 1. 2 j j − 1 j=1 It follows from Stirling’s formula that 2k 22k ≤ √ k 2k for all k ∈ N. Thus, we obtain that m m+1 X m m X m + 1 m + 1 2−m − + 2 = 2−(m+1) − + 2 j j − 1 j j − 1 j=1 j=1 2k 2 2 = 2−m ≤ √ ≤ √ , k m + 1 m which completes the proof.

91 Appendix

Proof of Theorem A.1. By Lemma A.2 (b), we have

S(t)|T (s)y| ≤ S(t)T (s)|y| ≤ S(t + s)|y| = 0

for all y ∈ E2 and t, s ∈ R. Thus, the closed ideal E2 is T -invariant. Since ﬁxs(T ) = ﬁx(T ) by Lemma 4.1.8, it follows from

T (t)(I − P )e = e − PT (t)P e ≥ e − PT (t)e = (I − P )e

for all t ∈ R that (I − P )e and hence also P e is a ﬁxed point of T . Deﬁne e0 := (I − P )e ∈ E0 and e2 := P e ∈ E2. As E0 is the closure of the principal ideal generated by e0, we conclude that also E0 is T -invariant.

It follows immediately from (A.1) that D(t)e2 = 2e2 for all t ∈ R. Hence, it remains to show that lim D(t)e0 = 0. For simplicity, we omit the index 0 from now

on and write E = E0, e = e0, T (t) = T (t)|E0 and so forth. Now, lim S(t)y > 0 for all y ∈ ﬁx(T ) \{0} by Lemma A.2 (b) and the deﬁnition of E0. Aiming for a contradiction, we assume that h := lim D(t)e > 0. Since h ≤ 2e, there exists m ∈ N such that

2 + k := h − √ e > 0. m

As fix(T ) is a sublattice by Lemma 4.1.8, we have k ∈ fix(T ). Therefore, Lemma m A.2 (c) asserts that S(t0) k > 0 for some t0 ∈ R. Let t1 := m(t0 + τ) and define the operator

m −m m U := T (t1) − 2 S(t0) I + T (τ) . (A.2)

It follows from S(t0)(I + T (τ)) ≤ T (t0 + τ) + T (t0)T (τ) = 2T (t0 + τ) that U is positive. Moreover,

m n −m T (nt1) = U + Rn2 I + T (τ) (A.3)

m n holds for all n ∈ N where R1 := S(t0) and Rn+1 := U R1 + RnT (t1). It follows n n from e = U e + Rne that 0 ≤ Rne ≤ e and 0 ≤ U e ≤ e for all n ∈ N. Now we

92 B. Axmann’s theorem conclude from Lemma A.2 (a) and Lemma A.3 that

h ≤ D(nt1)e = |T (nt1)(I − T (τ))|e m X m ≤ 2U ne + R 2−m T j(τ)(I − T (τ)) e n j j=0 m m+1 X m X m = 2U ne + R 2−m T j(τ) − T j(τ) e n j j − 1 j=0 j=1 m X m m = 2U ne + R 2−m 2e + − e n j j − 1 j=1 2 2 ≤ 2U ne + R √ e ≤ 2U ne + √ e n m m

N : n √2 for every n ∈ . Let y = limn→∞ U e ≥ 0. Then h ≤ 2y + m e and hence

2 + 0 < k = h − √ e ≤ 2y. m

Since y is a fixed point of U, it follows from equation (A.3) that T (nt1)y ≥ y. As s fix (T ) = fix(T ), this implies that T (nt1)y = y for every n ∈ N. By equation (A.2) we have m m m 0 = S(t0) I + T (τ) y ≥ S(t0) y ≥ 0.

m m Therefore, 0 < S(t0) k ≤ 2S(t0) y = 0 which contradicts our foregoing observa- m tion that S(t0) k > 0. Hence, h = lim D(t)e = 0 as desired.

B Axmann’s theorem

We give a proof Axmann’s theorem from [14, Satz 3.5] stating that an irreducible Harris operator has non-disjoint powers. A proof of this result for E = Lp can be found in [9, Sec 6]. We ﬁrst give a proof in the case where E is an L-space and reduce the general case to it in what follows. Let us recall that a Banach lattice E is called an L-space if kx + yk= kxk + kyk holds for all x, y ∈ E+. Proposition B.1. Let T be a positive and irreducible operator on an L-space E such that T 6∈ (E∗ ⊗ E)⊥. Then there exists n ∈ N, n ≥ 2, such that T ∧ T n > 0.

93 Appendix

Proof. Aiming for a contradiction, we assume that T ∧ T n = 0 for all n ≥ 2. Since E is an L-space, we may identify E∗ with C(K) for some compact space K by Kakutani’s theorem [49, Thm 2.1.3]. For n, m ∈ N, n ≥ 2, deﬁne

∗ ∗n An := {T h + T (1 − h): h ∈ C(K), 0 ≤ h ≤ 1} ⊆ C(K)

and On,m := {t ∈ K : h(t) < 1/m for some h ∈ An}. It follows from our assumption and Synnatschke’s theorem [49, Prop. 1.4.17] that

∗ ∗n n ∗ inf An = (T ∧ T )1 = (T ∧ T ) 1 = 0

for all n ≥ 2. Now, we show that each open set On,m is dense in K. Aiming for a contradiction, we assume the contrary. Then there exists a non-empty open set U ⊆ K\On,m for some n, m ∈ N. By Urysohn’s theorem, we can construct a 1 1 continuous function g : K → [0, m ] vanishing on On,m such that g(t0) = m for some t0 ∈ U. Hence, g > 0 is a lower bound of An in contradiction to inf An = 0. Therefore, every On,m and hence, by Baire’s theorem, also G := ∩n,mOn,m is dense in K. Note that for all t ∈ G and n ≥ 2 we have that

∗∗ ∗∗ n hT δt ∧ (T ) δt, 1i = inf{h(t): h ∈ An} = 0

∗∗ ∗∗ n and therefore T δt ∧ (T ) δt = 0. ∗ ⊥ ∗ ∗ Since T 6∈ (E ⊗ E) , there exist x ∈ E+ and y ∈ E+ such that T is not disjoint from R = y∗ ⊗ x. Then R∗ = x ⊗ y∗ corresponds to a rank-one operator ∗ ∗ ∗ ∗ µ ⊗ g on C(K) for some µ ∈ C(K)+ and g ∈ C(K)+. As R ∧ T ≥ (R ∧ T ) > 0, ∗ ∗ there exists some e ∈ C(K)+ such that % := (R ∧ T )e > 0. For all 0 ≤ h ≤ e and t ∈ K it follows from

% = (R∗ ∧ T ∗)h + (R∗ ∧ T ∗)(e − h) ≤ R∗h + T ∗(e − h)

that ∗ ∗∗ ∗∗ %(t) ≤ hR h, δti + he − h, T δti = g(t)hµ, hi + he − h, T δti. ∗∗ Taking the infimum over all 0 ≤ h ≤ e shows that %(t) ≤ (g(t)µ ∧ T δt)e for all t ∈ K. Now fix t ∈ {s ∈ K : %(s) > 0} ∩ G, which exists since G is dense in K. ∗∗ Then ν := g(t)µ ∧ T δt > 0 because %(t) > 0. As ν is dominated by g(t)µ and µ corresponds to x ∈ E, ν itself corresponds to a vector v ∈ E, v > 0, since E is an ideal in E∗∗. This vector v satisfies

n ∗∗ ∗∗ n+1 v ∧ T v ≤ T δt ∧ (T ) δt = 0

for all n ∈ N because t ∈ G.

94 B. Axmann’s theorem

Finally, we consider w := T v. If w = 0, then the closed ideal Ev is T -invariant and non-trivial since T 6= 0. If w > 0, then the closure of the T -invariant ideal

J := {z ∈ E : |z| ≤ c(w + T w + ··· + T kw) for some c > 0 and m ∈ N} is non-trivial because w ∈ J and v ∈ J ⊥. In both cases, T cannot be irreducible. Thus, we conclude that T ∧ T n > 0 for some n ≥ 2.

Theorem B.2. Let T be a positive and irreducible operator on E, a Banach lattice with order continuous norm, such that T 6∈ (E∗ ⊗ E)⊥. Then there exists n ∈ N, n ≥ 2, such that T ∧ T n > 0.

∗ ∗ ∗ ∗ −1 ∗ Proof. Fix λ > kT k and y ∈ E+ \{0}. Then z := (λ − T ) y satisﬁes λz∗ − T ∗z∗ = y∗ and hence T ∗z∗ ≤ λz∗. This implies that the null ideal N(z∗) = {x ∈ E : h|x|, z∗i = 0} is T -invariant and thus N(z∗) = {0}. Therefore, ∗ kxkz∗ := hz , |x|i deﬁnes an order continuous lattice norm on E. Let (F, k · kF ) ∗∗ be the completion of (E, k · kz∗ ), i.e. the closure of E in (E, k · kz∗ ) . Then F is an L-space and, since kT xkF ≤ λkT kF for all x ∈ E, T uniquely extends to a positive operator T˜ on F . Now, it follows as in the proof of Proposition 4.3.8 that T˜ is irreducible and T˜ 6∈ (F ∗ ⊗ F )⊥. Since the norm on E is order continuous, so is z∗ and hence E is an ideal in F by [55, Lem IV 9.3]. Thus, for x ∈ E+ and n ∈ N we have that

(T˜ ∧ T˜n)x = inf{T˜(x − y) + T˜ny : y ∈ F, 0 ≤ y ≤ x} F = inf{T (x − y) + T ny : y ∈ E, 0 ≤ y ≤ x} E = (T ∧ T n)x.

n n As E+ is dense in F+, this shows that T˜ ∧ T˜ = 0 if and only if T ∧ T = 0. Now the assertion follows from Proposition B.1.

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101

Index

L-space, 9 extremely disconnected, 9 M-space, 9 σ-adjoint, 15 Harris operator, 50

Abel averages, 29 ideal, 8 atom, 50 kernel atomic, 50 transition, 16 average scheme, 28 kernel operator, 49 (eventual) e-property, 37 Markovian, 39 lattice, 7 weakly ergodic, 31 lattice isomorphism, 10

Banach lattice, 8 mean ergodic theorem, 25 band, 8 band projection, 10 norm strictly monotone, 51 carrier, 55 norming dual pair, 14 Cesàroaverages, 25, 26 null ideal, 55 clustering net, 31 complex Banach lattice, 11 operator complexification, 61 τ-equicontinuous, 39 consistent topology, 15 Abel bounded, 29, 62 countable sup property, 54 Cesàrobounded, 29 countably order continuous, 10 Dedekind complete, 8 expanding, 68 diffuse, 50 Feller, 19 directed set, 9 integral, 48 disjoint complement, 8 irreducible, 10 disjoint elements, 8 kernel, 49 Markovian, 39 ergodic projection, 31 mean ergodic, 25

103 Index

non-disjoint powers, 72 positive, 11 order bounded, 10 strongly continuous, 11 order continuous, 10 time-continuous, 11 positive, 10 time-discrete, 11 power-bounded, 25, 29 star convergence, 54 regular, 10 star–order continuous, 56 strictly positive, 63 Stonian, 9 strong Feller, 38 strict topology, 15 strongly positive, 68 strictly positive functional, 54 ultra Feller, 38 strong unit, 9 weakly continuous, 18 super ﬁxed point, 51 order bounded, 7 order complete, 8 theorem of order continuous norm, 9 Bukhvalov, 53 order convergence, 10 Jacobs, de Leeuw, Glicksberg, 63 partial order, 7 Kakutani, 58 positive, 8 topology cone, 8 consistent, 15 projection band, 8 weak, 15 transition kernel, 16 quasi-interior point, 9 associated, 18

Riesz space, 8 uniformly tight, 39 unit, 9 s-o-continuous, 56 semigroup, 11 vector lattice, 8 Bb-ergodic, 83 normed, 8 (eventual) e-property, 37 Abel bounded, 62 weak topology, 15 Abelian, 11 weak unit, 9 adjoint, 28 weakly continuous operator, 18 bounded, 11 zero-two law, 89 Ces`arobounded, 30 contractive, 11 ﬁxed space, 26 generator, 11 integrable, 30 irreducible, 11 Markovian, 39 mean ergodic, 26 non-disjoint operators, 72

104 105

Nomenclature

(T (t))t≥0 Time-continuous semigroup

(T (t))t∈R Time-discrete or time-continuous semigroup (X,Y ) Norming dual pair

β0 Strict topology on Cb(Ω) B(Ω) Borel σ-algebra on Ω C (K) Closed and open subsets of K

L (E)+ Positive operators on E L (X) Bounded linear operators on X L (X, τ) τ-continuous operators on X L r(E) Regular operators on E L r(E,F ) Regular operators from E to F

r Lcoc(E,F ) Countably order continuous operators from E to F r Loc(E,F ) Order continuous operators from E to F

Lσ(E) Bounded operators on E endowed with the weak operator topology

L?o Star–order continuous operators M (Ω) Signed measures on Ω P(Ω) Power set of Ω ﬁx(S ) Fixed space of semigroup S ﬁx(T ) Fixed space of operator T

107 Nomenclature

ﬁxs(S ) Set of super ﬁxed points of semigroup S Γ Unit circle {z ∈ C : |z| = 1}

λ, µ, ν Measures

span M Linear hull of set M

Lipb(Ω, d) Bounded Lipschitz continuous functions on Ω w.r.t. metric d o-lim Order limit

σ Weak topology σ(X,Y )

σ0 Weak topology σ(Y,X)

0 σb The σ(Bb(Ω), M (Ω)) topology on Bb(Ω) 0 σc The σ(Cb(Ω), M (Ω)) topology on Cb(Ω)

σb The σ(M (Ω),Bb(Ω)) topology on M (Ω)

σc The σ(M (Ω),Cb(Ω)) topology on M (Ω) ?-lim Star limit

Aα Average scheme

Bb(Ω) Bounded Borel-measurable functions on Ω C(ϕ) Carrier of functional ϕ

Cb(Ω) Bounded continuous functions on Ω E, F Banach lattices

∗ Ecoc Countably order continuous functionals on E

∗ Eoc Order continuous functionals on E

EC Complexiﬁcation of Banach lattice E

Ex Principal ideal generated by x in E k( · , · ) Transition kernel

N(ϕ) Null ideal of functional ϕ

R(λ, A) Resolvent of operator A

108 T 0 σ-adjoint of T T ∗ Norm adjoint of T

TC Complexiﬁcation of operator T X, Y Banach spaces

109

Zusammenfassung

In der vorliegenden Arbeit studieren wir positive Halbgruppen aus Kernoperatoren hinsichtlich folgender Aspekte. Zum einen stellen wir abstrakte Charakterisie- rungen von Kernoperatoren vor, mit denen wir insbesondere Kernoperatoren auf Räumen von Maßen identifizieren. Zum anderen untersuchen wir das asymptotische Verhalten von Halbgruppen aus Kernoperatoren. Von besonderem Interesse dabei sind hinreichende Bedingungen fur¨ Mittelergodizität, das heißt Konver- genz der Halbgruppe im Mittel, sowie fur¨ Stabilität, das heißt Konvergenz der Halbgruppe selbst. Ein Operator auf einem Lp-Raum heißt Kernoperator, wenn er von der Form Z T f = h( · , y)f(y) dy fur¨ eine messbare Funktion h ist. Fur¨ einen regulären und ordnungsstetigen Operator ist dies genau dann der Fall, wenn er in dem Band liegt, das von den Operatoren endlichen Ranges erzeugt wird. Diese Charakterisierung dient als Definition von Kernoperatoren auf einem beliebigen Banachverband. Ferner können Kernoperatoren durch eine Stetigkeitsbedingung beschrieben werden. Bukhvalov zeigte in [16], dass ein Operator auf einem Lp-Raum genau dann ein Kernoperator ist, wenn er ordnungsbeschränkte normkonvergente Folgen auf fast uberall¨ konvergente abbildet. Wir verallgemeinern dieses Kriterium auf Operatoren auf Banachverbänden und charakterisieren abstrakte Kernoperatoren durch eine entsprechende Stetigkeitsbedingung. Diese Charakterisierung stammt ursprunglich¨ von Grobler und van Eldik [35]. Wir geben in Abschnitt 4.2 einen alternativen Beweis unter anderen Bedingungen an die zugrunde liegenden Räume. Viele Ergebnisse uber¨ das asymptotische Verhalten von Halbgruppen stellen eine Verbindung zu spektralen Eigenschaften her. Die in dieser Arbeit verwendete Technik ist davon grundsätzlich verschieden und geht auf Greiner zuruck.¨ Dieser gibt in [33] zwei hinreichende Bedingungen an, unter denen eine positive, kontrakti- ve und irreduzible stark stetige Halbgruppe auf Lp gegen eine Projektion auf ihren

111 Zusammenfassung

Fixraum konvergiert: Das periphere Punktspektrum ihres Generators ist {0} – insbesondere besitzt die Halbgruppe einen Fixpunkt – und die Halbgruppe enthält zwei nicht-disjunkte Operatoren. Aufgrund eines Satzes von Axmann ist letztere Bedingung automatisch erfullt,¨ wenn die Operatoren der Halbgruppe Kernope- ratoren dominieren. Solche Operatoren werden auch Harrisoperatoren genannt. Greiner bewies, dass die erste Bedingung bei Halbgruppen aus Kernoperatoren stets erfullt¨ ist und erhielt so einen Stabilitätssatz fur¨ diese Halbgruppen. Wir verallgemeinern Greiners Ergebnis in Abschnitt 5.1 in mehrerlei Hinsicht. Zum einen betrachten wir auch zeitdiskrete Halbgruppen und allgemeine Ba- nachverbände. Zum anderen zeigen wir, dass das periphere Punktspektrum von Generatoren von Halbgruppen aus Harrisoperatoren stets trivial ist. Dadurch erhalten wir einen Stabilitätssatz fur¨ diese Halbgruppen.

Daruber¨ hinaus erschließen wir eine andere aber verwandte Klasse positiver Halbgruppen fur¨ die Anwendung dieser Technik. Die Theorie der Markovprozesse befasst sich mit der zeitlichen Entwicklung von Wahrscheinlichkeitsverteilungen, die durch Halbgruppen gewisser positiver Operatoren auf Räumen von Maßen beschrieben wird. Diese Operatoren werden in der Literatur ebenfalls Kernopera- toren genannt, da sie durch Integrale uber¨ sogenannte Ubergangskerne¨ gegeben sind. Weil es sich andererseits um genau diejenigen positiven Operatoren handelt, die bezuglich¨ der von den beschränkten messbaren Funktionen auf dem Raum der Maße induzierten Topologie stetig sind, werden sie in dieser Arbeit als schwach stetige Operatoren bezeichnet, um sie von obigen Kernoperatoren zu unterscheiden. In Kapitel 2 untersuchen wir schwach stetige Operatoren erstmals systematisch innerhalb der Verbandstheorie. Dabei stellen wir Unterschiede und Gemeinsamkei- ten von schwach stetigen Operatoren und Kernoperatoren fest. Wir zeigen, dass die Menge der schwach stetigen Operatoren einen abzählbar ordnungsvollständigen Unterverband der regulären Operatoren darstellt, jedoch – im Gegensatz zu der Menge der Kernoperatoren – kein Ideal bildet. Diese Verbandsstruktur erlaubt uns, Greiners Technik auf gewisse Halbgruppen schwach stetiger Operatoren anzuwenden, wodurch wir einen rein analytischen Beweis des als Satz von Doob bekannten Stabilitätsresultats erhalten.

Eine notwendige Bedingung fur¨ Stabilität einer Halbgruppe ist Mittelergo- dizität, das heißt Konvergenz der Halbgruppe im Mittel. Ob eine Halbgruppe mittelergodisch ist, kann typischerweise leicht mit dem bekannten Mittelergoden- satz uberpr¨ uft¨ werden. Unter anderem charakterisiert dieser Mittelergodizität einer Halbgruppe dadurch, dass der Fixraum der Halbgruppe den Fixraum der adjungierten Halbgruppe trennt. Im Fall von Markovprozessen ist dieses Kriterium jedoch schwerlich anzuwenden, weil der Dualraum der Maße bezuglich¨ der Normtopologie zu groß ist, als dass der Fixraum der adjungierten Halbgruppe effektiv bestimmt werden kann.

112 Außerdem ist es häufig naturlicher,¨ Konvergenz der Mittel bezuglich¨ der von den beschränkten stetigen Funktionen induzierten Topologie, der sogenannten schwachen Topologie, zu studieren. In Kapitel 3 charakterisieren wir Mittelergodizität in dieser schwachen Topologie im Stil des klassischen Mittelergodensatzes dadurch, dass der Fixraum der Halbgruppe den Fixraum der adjungierten Halbgruppe auf den stetigen Funktionen trennt. Während Mittelergodizität im Allgemeinen schwächer ist als Konvergenz der Halbgruppe selbst, ist eine klassische Frage der Analysis, unter welchen Bedingungen die Konvergenz der Mittel bereits die Konvergenz der Halbgruppe impliziert. Aussagen dieses Typs werden auch als Tauber’sche Sätze bezeichnet. Wir sagen, dass ein beschränkter Operator auf einem Raum signierter Maße die starke Fellereigenschaft hat, falls seine Adjungierte beschränkte messbare Funktionen auf stetige abbildet. In Abschnitt 5.3 zeigen wir, dass fur¨ Markov- halbgruppen mit dieser Eigenschaft Mittelergodizität in der schwachen Topologie schon die Konvergenz der Halbgruppe selbst impliziert. Dieses Ergebnis erlaubt es, im Gegensatz zum Satz von Doob, auf eine Irreduzibilitätsvoraussetzung zu verzichten und Konvergenz der Halbgruppe auch im Fall eines Fixraums beliebig hoher Dimension nachzuweisen.

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Lebenslauf

Der Lebenslauf ist in der Online-Version aus Grunden¨ des Datenschutzes nicht enthalten.

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