Semigroups of Kernel Operators

Transcription

1 Semigroups of Kernel Operators Moritz Gerlach Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Mathematik und Wirtschaftswissenschaften der Universität Ulm. Vorgelegt von Moritz Gerlach aus Limburg an der Lahn im Jahr 2014.

2 Tag der Prüfung: 28. Mai 2014 Gutachter: Prof. Dr. Wolfgang Arendt Prof. Dr. Günther Palm Prof. Dr. Rainer Nagel Amtierender Dekan: Prof. Dr. Dieter Rautenbach

3 Contents Introduction 1 1 Preliminaries 7 2 Weakly continuous operators on the space of measures Norming dual pairs The lattice of transition kernels The sublattice of weakly continuous operators Mean ergodic theorems on norming dual pairs Average schemes An ergodic theorem on norming dual pairs An ergodic theorem on the space of measures Counterexamples Kernel and Harris operators Definition and basic properties Characterization by star order continuity Triviality of the peripheral point spectrum Stability of semigroups Stability of semigroups of Harris operators Doob s theorem A Tauberian theorem for strong Feller semigroups Appendix 89 Bibliography 97 Index 103

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5 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) = t 2 u(t, x) x t 0, x R u(0, x) = u 0 (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 u 0 at time t = 0. It is well-known that this equation has a unique bounded solution for every continuous and bounded initial value u 0. This solution is given by (T (t)u 0 )(x) = 1 4πt exp ( ) (x y)2 u 0 (y) dy. 4t The so defined operators T (t), which map an initial value u 0 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)u 0 instead of u 0. 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 T (t)f = h t (, y)f(y) dy 1

6 Introduction 2 for some function h t. 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 h t 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 L p 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 L p 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

7 showed that a positive, contractive and irreducible strongly continuous semigroup on L p 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 L p 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 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 T 1 T 2 of two weakly continuous operators is again weakly continuous and given by the infimum of their transition kernels k 1 and k 2 defined pointwise as the infimum of the measures k 1 (x, ) and k 2 (x, ). 3

8 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. 4 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 L p -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), C b (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 C b (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), C b (Ω)). Then an additional assumption, weaker than

9 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 timediscrete 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), C b (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

10 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 Glück 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, Information and Complexity for the financial support in the years 2010 and

11 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, (c) x y and y z implies x z. We also write x y for y x and x < y shorthand for x y and x 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

12 1. Preliminaries 8 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 x y 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 L p (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 L p (Ω, Σ, µ) is order complete.

13 If Ω is a topological space, the bounded Borel measurable functions B b (Ω) and the bounded continuous functions C b (Ω), both endowed with the uniform norm, form a Banach lattice with respect to the pointwise ordering. These spaces are in general not order complete. For a compact topological space K, C b (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 and see [3, Sec 10.11]. (µ ν)(b) = sup{µ(a) + ν(b \ A) : A B, A Σ} (µ ν)(b) = inf{µ(a) + ν(b \ A) : A B, A Σ}, 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{ x : 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 x + y = x + y for all x, y E +. If x y = max{ x, y } 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 E x 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 L p (Ω, Σ, µ) is order continuous for 1 p <, where L 1 (Ω, Σ, µ) is even an L-space. The Banach lattice of signed measures on (Ω, Σ) is also an L-space and L, B b and C b 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

14 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 := P E 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} for all x E + and T, S L r (E, F ) [49, Thm 1.3.2]. We write L r (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 (x α ) α Λ in E is called order convergent to x if there exists a decreasing net (z α ) α Λ with inf z α = 0 such that x α x z α for every index α Λ; in symbols o-lim x α = x. An operator T L r (E, F ) is called order continuous if o-lim T x α = T x for every net (x α ) E that order converges to x. If o-lim T x n = T x for every sequence (x n ) E that order converges to x, then T is called countably order continuous. We write L r (E, F ) oc, L r (E, F ) coc, E oc and E coc 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 E C := E ie of E there exists z := sup{cos(ϕ)x + sin(ϕ)y : 0 ϕ 2π} 10

15 and satisfies the following conditions. (a) z = 0 if and only if z = 0. (b) λz = λ z for all λ C and z E C. (c) z + w z + w for all z, w E C. If we endow E C with the norm z C := z, then E C is a Banach space which is called a complex Banach lattice. Every linear operator T : E F uniquely extends to a C-linear operator T C : E C F C defined as T C (x + iy) = T x + it y. If T is regular and F is order complete, then T C z T z for all z E C [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 = T S for all S, T S, then S is called Abelian. We say that a semigroup S L (X) is time-discrete if S = {T n : n N 0 } 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 lim t 0+ T (t)x = x for all x X is said to be strongly continuous. A semigroup is called bounded if sup T S T < and contractive if sup T S T 1. For a strongly continuous semigroup S = (T (t)) t 0 on a Banach space X, the operator A: D(A) X defined as on its domain Ax := lim (T (h)x x) h h 0 1 D(A) := {x X : lim h 0 (T (h)x x) exists} 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 a time-discrete semigroup S = {T n : n N 0 } is irreducible if and only if T is irreducible, whereas in general an irreducible semigroup does not contain an irreducible operator, see [33]. 11

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17 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 B b (Ω). Such operators are called weakly continuous and are characterized by the condition that their adjoints leave the space B b (Ω) invariant, i.e. the operators act on the dual pair (M (Ω), B b (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 B b (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), σ b ), the space of all weakly continuous operators. First, in Section 2.2, we recall that each T L (M (Ω), σ b ) is of the form 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

18 2. Weakly continuous operators on the space of measures 14 over to L (M (Ω), σ b ). Thus, for every T L (M (Ω), σ b ) there exists a least upper bound of {T, T } inside L (M (Ω), σ b ). On the other hand, the modulus of each T L (M (Ω), σ b ) 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 A norming dual pair is a triple (X, Y,, ) where X and Y are Banach spaces and, is a duality between X and Y such that x = sup{ x, y : y Y, y 1} and y = sup{ x, y : x X, x 1}. Identifying y with the linear functional x x, y, 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, 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,. If (Ω, Σ) is a measurable space, we write B b (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 (B b (Ω), M (Ω)) is a norming dual pair with respect to the duality f, µ := f dµ. 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 (C b (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. Ω

19 2.2. The lattice of transition kernels Definition Let (X, Y ) be a norming dual pair. We call a locally convex topology τ on X consistent if (X, τ) = Y, i.e. every τ-continuous linear functional ϕ on X is of the form ϕ(x) = x, y 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 σ for the σ(y, X) topology on Y. To indicate convergence with respect to σ and σ, we write σ- lim and σ - lim, respectively. If the norming dual pair under consideration is (M (Ω), B b (Ω)) for a Polish space Ω, we denote the weak topologies by σ b := σ(m (Ω), B b (Ω)) and σ b := σ (B b (Ω), M (Ω)). For the norming dual pair (M (Ω), C b (Ω)) we write σ c and σ c shorthand for σ(m (Ω), C b (Ω)) and σ (C b (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 An operator T L (X) belongs to L (X, σ) if and only if its norm adjoint T leaves Y invariant. In that case, T X = T Y Y. For a σ-continuous operator T L (X, σ) we write T for the σ-adjoint T Y. It follows easily from Lemma that L (X, σ) is a subalgebra of L (X) which is closed in the operator norm. For the norming dual pair (C b (Ω), M (Ω)) on a Polish space Ω also the strict topology β 0 is important. Definition Let F 0 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 β 0 on C b (Ω) is the locally convex topology generated by the set of seminorms {p ϕ : ϕ F 0 } where p ϕ (f) := ϕf. The strict topology is consistent with the duality, i.e. (C b (Ω), β 0 ) = M (Ω), see [38, Thm 7.6.3], and it coincides with the compact open topology on norm bounded subsets of C b (Ω) [38, Thm ]. Moreover, it is the Mackey topology of the dual pair (C b (Ω), M (Ω)) [57, Thm 4.5, 5.8], i.e. it is the finest locally convex topology on C b (Ω) which yields M (Ω) as a dual space. In particular, L (C b (Ω), σ c) = L (C b (Ω), β 0 ), 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), B b (Ω)) and prove that the space 15

20 2. Weakly continuous operators on the space of measures 16 L (M (Ω), σ b ) 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 A transition kernel on Ω is a map k : Ω B(Ω) R with the following properties: (a) A k(x, A) is a signed measure for every x Ω and (b) x 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 sup x Ω k (x, Ω) <. To each bounded transition kernel k, one may associate two mutually adjoint operators T L (M (Ω)) and S L (B b (Ω)) in the following way. Lemma Let k be a bounded transition kernel. By setting (T µ)(a) := k(x, A) dµ(x) (2.2.1) for µ M (Ω) and A B(Ω) and (Sf)(x) := Ω Ω f(y)k(x, dy) (2.2.2) for f B b (Ω) and x Ω, we obtain operators T L (M (Ω)) and S L (B b (Ω)) satisfying T Bb (Ω) = S and S M (Ω) = T. Moreover, T = S = 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 = 1 A, where A B(Ω), we have that Sf = k(, A) is a bounded measurable function by definition of k. Thus, by linearity, Sf B b (Ω) for every measurable simple function f. Approximating an arbitrary f B b (Ω) uniformly by a sequence of simple functions (f n ) satisfying f n f, we obtain that Sf is measurable as the pointwise limit of Sf n and (Sf)(x) f(y) k (x, dy) f C Ω

21 2.2. The lattice of transition kernels for all x Ω. This shows that S L (B b (Ω)) with S C. For a finite partition Z B(Ω) of Ω and x Ω we define f x,z := A Z sign k(x, A) 1 A and obtain that S (Sf x,z )(x) = A Z k(x, A). Taking the supremum over all finite partitions Z of Ω and all x Ω implies that S C. Now we turn to the operator T. Let µ M (Ω) and let (A n ) B(Ω) be a sequence of pairwise disjoint Borel sets. By definition of k (T µ)( n N A n ) = Ω k(x, n N A n ) dµ(x) = k(x, A n ) dµ(x) Ω n=1 for all x Ω. Since N k(x, A n ) = k(x, N n=1a n ) C n=1 for all x Ω and N N and since the constant C is µ-integrable, k(x, A n ) dµ(x) = Ω n=1 n=1 Ω k(x, A n ) dµ(x) = (T µ)(a n ) by the dominated convergence theorem. Thus, T µ is σ-additive and therefore a measure. It follows immediately that T µ, 1 A = µ, S1 A for every A B(Ω). Hence, by linearity, T µ, f = µ, Sf for every measurable simple function. Approximating f B b (Ω) uniformly by a sequence (f n ) B b (Ω) of simple functions shows that T µ, f = µ, Sf holds for all f B b (Ω). Thus, T Bb = S and S M (Ω) = T. The identity T = S now follows from Lemma Lemma is well-known in the literature, see e.g. [45, Lem 2.2.2]. Combining it with Lemma we see that the operator given by (2.2.1) is σ b -continuous and the operator given by (2.2.2) σ b -continuous. The following Lemma shows that σ b -continuity characterizes operators of this form. n=1 Lemma 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

22 2. Weakly continuous operators on the space of measures 18 (ii) The norm adjoint T of T leaves B b (Ω) invariant. (iii) The operator T is continuous in the σ b = σ(m (Ω), B b (Ω)) topology. Proof. Lemma shows that (i) implies (ii) which is equivalent to (iii) by Lemma If (ii) holds, then S := T Bb (Ω) is σ b -continuous by Lemma 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 Likewise, for an operator S L (B b (Ω)) it follows from Lemma that S is σ b -continuous if and only if S leaves the space M (Ω) invariant. Thus, S L (B b (Ω), σ b ) if and only if S satisfies the equivalent conditions of Lemma if and only if S = S Bb (Ω) is given by (2.2.2) for a bounded transition kernel k. Definition An operator T L (M (Ω)) is called weakly continuous if it satisfies the equivalent conditions in Lemma 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), σ b ) for the space of all weakly continuous operators. We order the transition kernels on Ω pointwise, i.e. k 1 k 2 if and only if k 1 (x, A) k 2 (x, A) for all x Ω and A B(Ω). With respect to this ordering the transition kernels form a lattice by Proposition below. Important for its proof is especially the fact that the lattice operations on C b (Ω) are β 0 -continuous. This follows from [54, V 7.1] since in the strict topology the origin has a neighborhood base of solid sets. Proposition 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 C b (Ω) is norming for M (Ω), we have that k(x, ) = k (x, Ω) = sup f C b (Ω) f 1 f, k(x, ) (2.2.3) 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 C b (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 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 k (x, A) is measurable}

23 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 C b (F ) such that for all measures µ M (F ), µ 0, there exists f M with µ, f 0. We denote by S := span Q M the linear span of M with rational coefficients. Now we show that the β 0 -closure of S, which is the same as the β 0 -closure of span M, is dense in C b (F ). Assume that there exists g C b (F ) which does not belong to the β 0 -closure of span M. Then, by the Hahn-Banach theorem, we find µ M (F ) such that µ, f = 0 for all f span M whereas µ, g 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 β 0 -dense in C b (F ) and the lattice operations are β 0 -continuous, D is β 0 -dense in the closed unit ball of C b (F ). Let x Ω and f C b (F ), f 1. Given ε > 0 we find g D such that g f, k(x, ) ε. Hence, From this it follows that f, k(x, ) g, k(x, ) + ε sup h, k(x, ) + ε h D sup h C b (F ) h 1 as desired. This finishes the proof. h, k(x, ) + ε = k (x, F ) + ε. k (x, F ) = sup f, k(x, ) f D Now let T 1, T 2 L (M (Ω)) be weakly continuous operators with associated transition kernels k 1, k 2. Noting that 1 A, T j δ x = k j (x, A) for j = 1, 2, we see that T 1 T 2 as operators on M (Ω) if and only if k 1 k 2 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 the following result. Theorem The space L (M (Ω), σ b ) is a lattice in its natural ordering inherited from L (M (Ω)). An operator T M (Ω) that is σ c = σ(m (Ω), C b (Ω))-continuous and thus acting on the norming dual pair (M (Ω), C b (Ω)) is often called a Feller operator in the literature. Such operators are in particular weakly continuous in the sense of Definition Indeed, by Lemma 2.1.3, T is a σ c = σ (C b (Ω), M (Ω))- continuous operator on C b (Ω) and thus given by (2.2.2) for a bounded transition 19

24 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 linear operator on B b (Ω), which we still denote by T. Since C b (Ω) separates the points in M (Ω), T = T M (Ω) is given by (2.2.1) and therefore is weakly continuous by Lemma 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 C b (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 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 n δ n+1 n N k(n, ) := 0 n ( N) { }. Then k (n, ) = { δ n + δ n+1 n N 0 n ( N) { }. Let T, U L (M (Ω), σ b ) denote the operators associated with k and k, respectively. Then T C b (Ω) C b (Ω). However, U maps the continuous function 1 Ω to the function 21 N which is not continuous. This shows that the σ(m (Ω), C b (Ω))- continuous operators do not form a sublattice of the σ(m (Ω), B b (Ω))-continuous operators. Moreover, there exists no modulus of T in the σ(m (Ω), C b (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 1 Ω (n) = 21 N (n) for n. But this shows that S 1 Ω cannot be continuous. This is a contradiction. 2.3 The sublattice of weakly continuous operators We still consider the norming dual pair (M (Ω), B b (Ω)) for a Polish space Ω endowed with its Borel σ-algebra. We have seen in Theorem that the space L (M (Ω), σ b ) of weakly continuous operators is a lattice, i.e. for each T L (M (Ω), σ b ) there exists a least upper bound of {T, T } inside L (M (Ω), σ b ), which is called the modulus of T. 20

25 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), σ b ) is neither an ideal nor order complete. In order to show that L (M (Ω), σ b ) is a sublattice of L (M (Ω)), we show that for T L (M (Ω), σ b ) 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 T + µ = sup T ν = 0 ν µ sup 0 ν µ = sup g B b (Ω) 0 g 1 for every positive measure µ, see [49, Thm 1.3.2]. Ω k(x, ) dν(x) Ω g(x)k(x, ) dµ(x) (2.3.1) Lemma Let k be a transition kernel, α > 0 and let U = {B n : n N} be a countable basis of the topology on Ω that is closed under finite unions. Then {k + (, Ω) > α} = {k(, B n ) > α}. n N Proof. If k(x, B n ) > α for some x Ω and n N, then clearly k + (x, Ω) k + (x, B n ) k(x, B n ) > 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 B n U with p(x, B n ) > α. Lemma Let T L (M (Ω), σ b ) with associated transition kernel k. Let α > 0 and A B(Ω) such that α1 A < k + (, Ω). Then (T + µ A )(Ω) αµ(a) where µ A denotes the measure µ(a ). 21

26 2. Weakly continuous operators on the space of measures 22 Proof. Let (B n ) n N be a countable basis of the topology on Ω that is closed under finite unions. We define E n := A {k(, B n ) > α} for n N. Then Lemma yields that A = A {k + (, Ω) > α} = E n. Defining Ω 1 := E 1 and Ω n := E n \ ( k<n E k ) for n > 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 ( ) µ Ω n µ(a) ε α. n N We now refine the sets B 1,..., B N 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 B n or disjoint from B n and (ii) we have B m = B n. m M We let N(m) := {n N : Bm B n } so that B n 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) Ω n, we find that (T + µ A )(Ω) M (T + µ A )( B m ) m=1 n N M m=1 n N n N(m) Ωn k(x, B m ) dµ(x). Since the sets Ω n as well as the sets B m are disjoint, we have that M m=1 n N(m) Ωn k(x, B m ) dµ(x) = As k(, B n ) > α on Ω n, we conclude that n=1 = = M m=1 n N(m) N n=1 m M n N(m) N n=1 n N Ω n k(x, B m ) dµ(x) Ω n k(x, B m ) dµ(x) Ω n k(x, B n ) dµ(x). N ( ) k(x, B n ) dµ(x) > αµ Ω n αµ(a) ε Ω n which completes the proof.

27 2.3. The sublattice of weakly continuous operators Theorem The L (M (Ω), σ b ) is a sublattice of L (M (Ω)). Proof. Let T L (M (Ω), σ b ) 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 f = M α j 1 Aj j=1 be a simple function with coefficients α j > 0 and pairwise disjoint sets A j B(Ω) such that f(x) < k + (x, Ω) for all x Ω and ( ) k + (x, Ω) f(x) dµ(x) < ε. Lemma yields that Ω M M (T + µ)(ω) (T + µ Aj )(Ω) α j µ(a j ) j=1 = Ω f(x) dµ(x) = (Sµ)(Ω) ε. j=1 Ω k + (x, Ω) dµ(x) ε Hence T + µ = Sµ and thus, since µ was arbitrary, T + = S L (M (Ω), σ b ). 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 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) \ B b (Ω) since ϕ, δ x = 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), σ b ) but T 1 = µ(ω)ϕ B b (Ω) and hence T L (M (Ω), σ b ). We conclude this section with an investigation of order completeness of the sublattice L (M (Ω), σ b ). We prove that this space is countably order complete but not order complete. Let us start with a well-known lemma. Lemma Let (µ n ) be an increasing sequence of positive measures on B(Ω) and ν M (Ω) such that µ n ν for all n N. Then µ := sup µ n is given by µ(a) = sup µ n (A) for all A B(Ω). 23

28 2. Weakly continuous operators on the space of measures Proof. Let f n denote the density of µ n with respect to ν and define f := sup f n. Then f dν = sup f n dν = sup µ n (A) n N n N A A for all A B(Ω) by the monotone convergence theorem. Therefore, the mapping A sup µ n (A) defines a measure on B(Ω) and thus (sup µ n )(A) = sup µ n (A) for all A B(Ω). Theorem Let (T n ) L (M (Ω), σ b ) be a sequence of weakly continuous operators that is order bounded by an element of L (M (Ω)). Then sup T n exists in L (M (Ω)) and is weakly continuous. Proof. Since M (Ω) is order complete, S := sup T n exists in L (M (Ω)). It remains to show that S L (M (Ω), σ b ). Note that by Theorem 2.3.3, T 1 T n is again weakly continuous. Thus, replacing T n by T 1 T n T 1 and S by S T 1, we may assume that (T n ) is increasing and T n 0 for all n N. We denote by k n the transition kernel associated with T n. For x Ω and A B(Ω) we define k(x, A) := sup k n (x, A) (Sδ x )(A) S. n N Then k(, A) is measurable for all A B(Ω) and k(x, ) is a measure by Lemma Hence, k is a bounded transition kernel. Since (T n ) is increasing, we have for all µ M (Ω) + that Sµ = (sup T n )µ = sup(t n µ) = sup n N k n (x, ) dµ(x) = k(x, ) dµ(x), where the last identity follows from Lemma and the monotone convergence theorem. The following example shows that L (M (Ω), σ b ) is not order complete, i.e. not every order bounded set has a supremum. Example 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 T x := 1 {x} µ. Then the set T := {T x : x E} is dominated by 1 µ. Let us assume that S := sup x E T x exists in L (M (Ω), σ b ). Fix x Ω \ E and denote by P the band projection onto {δ x }. Since P f = f 1 Ω\{x} B b (Ω) for all f B b (Ω), P is σ b -continuous by Lemma Thus, SP L (M (Ω), σ b ) is an upper bound of T and hence (Sδ x )(Ω) (SP δ x )(Ω) = 0. On the other hand, for y E we have (Sδ y )(Ω) (T y δ y )(Ω) = 1. Therefore, S 1 is not a measurable function. This contradicts our assumption that S L (M (Ω), σ). Ω Ω 24