Continuity and equicontinuity of semigroups on norming dual pairs

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C _b (E) which is continuous with respect to the strict topology β ₀ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β ₀-continuous semigroups on C _b (E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β ₀-continuous.     

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.     

On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C ₀ semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.      

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.     

On metrizable enveloping semigroups

When a topological group G acts on a compact space X, its enveloping semigroup E(X) is the closure of the set of g-translations, g ∈ G, in the compact space X ^X . Assume that X is metrizable. It has recently been shown by the first two authors that the following conditions are equivalent: (1) X is hereditarily almost equicontinuous; (2) X is hereditarily nonsensitive; (3) for any compatible metric d on X the metric d _G (x, y) ≔ sup{d(gx, gy): g ∈ G} defines a separable topology on X; (4) the dynamical system (G, X) admits a proper representation on an Asplund Banach space. We prove that these conditions are also equivalent to the following: the enveloping semigroup E(X) is metrizable.     

Group topologies coarser than the Isbell topology

The Isbell, compact-open and point-open topologies on the set C(X,R) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α(X) of compact families of open subsets of a topological space X. Those α(X) for which addition is jointly continuous at the zero function in Cα(X,R) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α(X) for which Cα(X,R) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, that Cα(X,R) can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.     

On the topology induced by the adjoint of a semigroup of operators

The adjoint of aC ₀-semigroup on a Banach spaceX induces a locally convex σ(X,X ^ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ^ℴ)-topology and it is shown that the uniform limit of σ(X,X ^ℴ)-compact operators is σ(X,X ^ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.     

Ergodic properties of operator semigroups in general weak topologies

The aim of this paper is to study ergodic properties (i.e., properties about the limit of Cesàro averages) of a semigroup of bounded linear operators on a Banach space X, which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of X. Then the results are applied to particular examples, such as locally strongly integrable semigroups, their dual semigroups, and the tensor product semigroup of two (C₀)-semigroups.     

A Hilbert cube compactification of the function space with the compact-open topology

Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)^ℕ of the Hilbert cube Q = [−1, 1]^ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $ \bar C $ \bar C (X) of C(X) such that the pair ($ \bar C $ \bar C (X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $ \bar C $ \bar C (X) coincides with the space USCC _F (X,      

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.     

2-Summing operators on C([0, 1], lp) with values in l1

Let Ω be a compact Hausdorff space, X a Banach space, C(Ω, X) the Banach space of continuous X-valued functions on Ω under the uniform norm, U: C(Ω, X) → Y a bounded linear operator and U ^#, U _# two natural operators associated to U. For each 1 ≤ s < ∞, let the conditions (α) U ∈ Π _s (C(Ω, X), Y); (β)U ^# ∈ Π _s (C(Ω), Π _s (X, Y)); (γ) U _# ε Π _s (X, Π _s (C(Ω), Y)). A general result, [10, 13], asserts that (α) implies (β) and (γ). In this paper, in case s = 2, we give necessary and sufficient conditions that natural operators on C([0, 1], l _p ) with values in l ₁ satisfies (α), (β) and (γ), which show that the above implication is the best possible result.     

Perturbations of Bi-continuous Semigroups with Applications to Transition Semigroups on C<Subscript>b</Subscript>(H)

We prove an unbounded perturbation theorem for bi-continuous semigroups on the space of bounded, continuous functions on the Hilbert space H. This is applied to the Ornstein-Uhlenbeck semigroup, thus providing a purely functional analytic approach to the existence of transition semigroups on C_b(H) with bounded non-linear drift.     

More stratifiable function spaces

If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M₁.     

Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L ¹(0, b, X), the convolution of T with f is defined by . It is shown that T * f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T * f is continuously differentiable for all f ∈ L ^p (0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L ¹(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators if one of them generates a C ₀-semigroup. Günter Lumer in memoriam