Montel Spaces Research Articles

Mean ergodic composition operators on spaces of smooth functions and distributions

We investigate (uniform) mean ergodicity of weighted composition operators on the space of smooth functions and the space of distributions, both over an open subset of the real line. Among other things, we prove that a composition operator with a real analytic diffeomorphic symbol is mean ergodic on the space of distributions if and only if it is periodic with period 2. Our results are based on a characterization of mean ergodicity in terms of Cesàro boundedness and a growth property of the orbits for operators on Montel spaces which is of independent interest.

Fréchet spaces with no infinite-dimensional Banach quotients

We exhibit examples of Fréchet Montel spaces E which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite-dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2. They are of independent interest and show for example that the canonical inclusion between James spaces J p ⊂ J q , 1 < p < q < ∞ , is strictly cosingular. This result requires a careful analysis of the block basic sequences of the canonical basis of the dual J p ′ of the James space J p , and permits us to show that the Fréchet space J p + = ⋂ q > p J q has no infinite-dimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no infinite-dimensional Banach subspaces.

Abelian groups which satisfy Pontryagin duality need not respect compactness

Let G be a topological Abelian group with character group GA. We will say that G respects compactness if its original topology and the weakest topology that makes each element of GA continuous produce the same compact subspaces. We show the existence of groups which satisfy Pontryagin duality and do not respect compactness, thus furnishing counterexamples to a result published by Venkataraman in 1975. Our counterexamples will be the additive groups of all reflexive infinite-dimensional real Banach spaces. In order to do so, we first characterize those locally convex reflexive real spaces whose additive groups respect compactness. They are exactly the Montel spaces. Finally, we study the class of those groups that satisfy Pontryagin duality and respect compactness. 0. INTRODUCTION AND NOTATION Let (G, z) be a topological Abelian group with underlying group G and topology T. A character of G is a homomorphism from G into the circle group T. Denote by (G, r)A the group of continuous characters of (G, T) with multiplication defined pointwise, equipped with the compact-open topology. We will say that (G, T) is a maximally almost periodic group if, for every a e G different from the identity, there exists X E (G, T)A such that X(a) $ 1. If (G, T) is indeed maximally almost periodic, we will say that (G, T) satisfies Pontryagin duality if the function t: (G, T) -* (G, T)AA, defined by t(x)(x) = x(x) for all x E G and X E (G, T)A, is a topological isomorphism. Let LCA, g, and MAP denote the classes of those groups that are locally compact, satisfy Pontryagin duality, and are maximally almost periodic, respectively. The classical theorem on duality due to Pontryagin and van Kampen states that LCA C p. The class p, however, is strictly wider than LCA: for example, 0 is closed under arbitrary products [6], a property that LCA does not hold. Let (G, T) e MAP. Denote by Tw the weakest topology on G that makes every element in (G, T)A continuous. It follows that (G, T,) is a totally bounded Received by the editors March 18, 1991 and, in revised form, July 20, 1991. 1991 Mathematics Subject Classification. Primary 22A05, 22D35, 46Al 1, 54A10; Secondary 46A20, 46A50. ? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

Abelian Groups which Satisfy Pontryagin Duality need not Respect Compactness

Let ${\mathbf {G}}$ be a topological Abelian group with character group ${{\mathbf {G}}^ \wedge }$. We will say that ${\mathbf {G}}$ respects compactness if its original topology and the weakest topology that makes each element of ${{\mathbf {G}}^ \wedge }$ continuous produce the same compact subspaces. We show the existence of groups which satisfy Pontryagin duality and do not respect compactness, thus furnishing counterexamples to a result published by Venkataraman in 1975. Our counterexamples will be the additive groups of all reflexive infinite-dimensional real Banach spaces. In order to do so, we first characterize those locally convex reflexive real spaces whose additive groups respect compactness. They are exactly the Montel spaces. Finally, we study the class of those groups that satisfy Pontryagin duality and respect compactness.

On Fréchet Montel spaces and their projective tensor product

AbstractThe main result in this note is as follows. Let E be a Fréchet Montel space which belongs to the large class of decomposable (FG)-spaces introduced by Bonet, Díaz, Taskinen and let F be a Fréchet Montel (resp. distinguished) space. Then the projective tensor product is Montel (resp. distinguished). We also give examples of Montel decomposable (FG)-spaces without an unconditional basis.

Operator algebras generated by Boolean algebras of projections in Montel spaces

It is shown that the classical result of W. Bade, stating that the uniformly closed operator algebra generated by a complete Boolean algebra of projections in a Banach space coincides with the weak operator closed algebra that it generates, is also valid (if suitably interpreted) in Montel spaces, Schwartz spaces and other “related” spaces.

The compact open and the Nachbin ported topologies on spaces of holomorphic functions

Some examples of Frechet Montel spaces E, which are not Schwartz spaces, for which the compact open and the Nachbin ported topologies coincide on the space of all holomorphic functions on an arbitrary balanced open subset of E, are given.

Weighted inductive and projective limits of normed Köthe function spaces

We consider locally convex inductive and projective limits of weighted normed Köthe function spaces of.measurable functions on a σ-finite measure space (X, Σ,μ). This general framework includes inductive and projective limits of weighted Lp-spaces, 1 ≤ p ≤ ∞, Lorentz spaces L p,q 1 < p < ∞, 1 ≤ q ≤ ∞, and Orlicz function spaces. In the projective case we characterize the quasinormability of L p(A) in terms of the increasing sequence A = (a n)n∈ℕ of weights on X. For the most important normed Köthe function spaces Lp, like Lp-spaces or L p,q-spaces, this condition on A = (an)n∈ℕ (V = (v n = 1/an)n∈ℕ) allows a useful projective description of the inductive limit topology of k p(V) by an associated system ofweights. A characterization of the Montel spaces L P(A) in terms of A = (an)n∈ℕ is also given. As an application we obtain functional analytic results on weighted inductive limits of spaces of holomorphic and harmonic functions.

On two new classes of locally convex spaces

The purpose of this paper is to introduce two new classes of locally convex spaces which contain the classes of semi-Montel and Montel spaces. Further we give some examples and study some properties of these classes. As to permanence properties, these classes have similar properties to semi-Montel and Montel spaces except strict inductive limits and these classes are not always preserved under their completions. We shall call these two classes as β–semi–Montel and β–Montel spaces. A β–semi–Montel space is obtained by replacing the word “bounded” by “strongly bounded” in the definition of a semi–Montel space. If a β–semi–Montel space is infra-barrelled, we call the space β–Montel.

Fréchet spaces with nuclear Köthe quotients

Each separable Fréchet non-Banach space $X$ with a continuous norm is shown to have a quotient $Y$ with a continuous norm and a basis. If, in addition, $Y$ can be chosen to be nuclear, we say that $X$ has a nuclear Köthe quotient. We obtain a (slightly technical) characterization of those separable Fréchet spaces with nuclear Köthe quotients. In particular, separable reflexive Fréchet spaces which are not Banach (and thus Fréchet Montel spaces) have nuclear Köthe quotients.

Holomorphic functions on strict inductive limits

We study strict inductive limits of Frechet Montel (FM) spaces and reflexive Frechet (RF) spaces and we obtain some interesting examples in the theory of infinite dimensional holomorphy.

On the Radon-Nikodým theorem and locally convex spaces with the Radon-Nikodým property

Let F be a quasi-complete locally convex space, ( Ω , Σ , μ ) (\Omega ,\Sigma ,\mu ) a complete probability space, and L 1 ( μ ; F ) {L^1}(\mu ;F) the space of all strongly integrable functions f : Ω → F f:\Omega \to F with the Egoroff property. If F is a Banach space, then the Radon-Nikodým theorem was proved by Rieffel. This result extends to Fréchet spaces. If F is dual nuclear, then the Lebesgue-Nikodým theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodým theorem is not available in general. It is shown in this article that the Radon-Nikodým theorem for the strong integral can be established for quasi-complete locally convex spaces F having the following property: (CM) For every bounded subset B ⊂ l N 1 { F } B \subset l_N^1\{ F\} , the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset M ⊂ F M \subset F such that Σ i = 1 ∞ p M ( x i ) &gt; 1 , ∀ ( x i ) ∈ B \Sigma _{i = 1}^\infty {p_M}({x_i}) &gt; 1,\forall ({x_i}) \in B . In fact, these spaces have the Radon-Nikodým property, and they include the Montel (DF)-spaces, the strong duals of metrizable Montel spaces, the strong duals of metrizable Schwartz spaces, and the precompact duals of separable metrizable spaces. When F is dual nuclear, the Radon-Nikodým theorem reduces to the Lebesgue-Nikodým theorem. An application to probability theory is considered.

The Lefschetz fixed point theorem for noncompact locally connected spaces

Leray's notion of convexoid space is localized and used to show that if f: M- M is a relatively compact map on a locally convex manifold M, and f has no fixed points then its Lefschetz trace is zero. A similar theorem holds for certain ad junction spaces Y Uj Z where Y is Q-simplicial and Z is locally convexoid. A number of other properties of locally convexoid spaces are derived; for example, any neighborhood retract of a locally convexoid space is locally convexoid. The Lefschetz fixed point theorem has long been known to hold for compact manifolds, including compact homological manifolds. By restricting attention to compact maps, Leray extended the Lefschetz theorem to open subsets of Banach spaces and more recently, Browder and Eells have obtained it for infinite dimen- sional manifolds modelled on Banach and Frechet spaces. The Leray, Browder and Eells results were each obtained by a reduction to the Lefschetz theorem for compact spaces. This same technique does not seem to apply to other infinite dimensional manifolds, such as those modelled on Montel spaces. In this paper we will consider this problem in the general context of regular Hausdorff locally connected spaces. For spaces which are locally connected in the sense of Cech, the Lefschetz fixed point theorem will be obtained for compact maps which have finite dimensional image. By localizing Leray's notion of convexoid space, and freeing it from compactness, it will be possible to obtain the Lefschetz theorem for compact maps of locally convex manifolds with no dimensional restrictions. Because of its applicability to noncompact spaces, the notion of Q?simplicial space will be used throughout, along with related techniques. As an auxiliary to these techniques, a relative form of the classical method of acyclic carriers, the method of acyclic pairs of interlaced carriers, will be applied- whenever it will be necessary to construct chain maps,

Schauder Bases and Kothe Sequence Spaces

In the light of the vast amount of literature on Schauder bases in Banach spaces which has appeared during the last thirty years, and considering the more recent development of the general theory of locally convex spaces, it is not surprising that many people are beginning to study Schauder bases in locally convex spaces. (See for example [1], [5], [7], [14], [16]-[21], [24].) One of the most important areas in the field of locally convex spaces is the theory of Kothe sequence spaces [10]. Not very much is gained, of course, by looking at the sequence space determined by a locally convex space with a Schauder basis. However, if one considers the case in which the topological dual and the Kothe dual coincide, then the two theories seem to come together very nicely. One obtains easy proofs of old theorems and important generalizations of others. Of course, as might be expected in any study involving Kothe sequence spaces, one obtains many counterexamples which mark out the boundaries of possible theorems. The authors feel that in this paper a new approach is introduced to both of the fields mentioned in the title and that although many new results are obtained, only the surface has been scratched. For example, no attempt has been made to study the specific types of locally convex spaces presently of interest to analysts, such as Frechet spaces, Q9,-spaces, Montel spaces, and nuclear spaces. It should be mentioned, however, that in almost all cases, the hypotheses of our general theorems are satisfied by these spaces. Aside from the overall approach, our main interest here is to study various types of Schauder bases which have previously been introduced, their relationships to each other and to various topological vector space concepts. In part I, we define the types of bases, interpret them in the context of a sequence space with a topology of uniform convergence on a family of subsets of the Kbthe dual and determine when a locally convex space can be viewed in this context. In part II we study the types of bases and their implications. The most important result is Theorem (1.10) which is a generalization of a theorem of R. C. James [9]. Finally in part III, we apply our results to the concepts of reflexivity and similar bases. Notation and terminology. If E[fT]; where E is a vector space over the field K of real or complex numbers and 9is a topology on E; is a locally convex space (always assumed to be Hausdorff) and (bn) is a sequence in E, we say that (bn) is a

Schauder bases and Köthe sequence spaces

In the light of the vast amount of literature on Schauder bases in Banach spaces which has appeared during the last thirty years, and considering the more recent development of the general theory of locally convex spaces, it is not surprising that many people are beginning to study Schauder bases in locally convex spaces. (See for example [1], [5], [7], [14], [16]-[21], [24].) One of the most important areas in the field of locally convex spaces is the theory of Kothe sequence spaces [10]. Not very much is gained, of course, by looking at the sequence space determined by a locally convex space with a Schauder basis. However, if one considers the case in which the topological dual and the Kothe dual coincide, then the two theories seem to come together very nicely. One obtains easy proofs of old theorems and important generalizations of others. Of course, as might be expected in any study involving Kothe sequence spaces, one obtains many counterexamples which mark out the boundaries of possible theorems. The authors feel that in this paper a new approach is introduced to both of the fields mentioned in the title and that although many new results are obtained, only the surface has been scratched. For example, no attempt has been made to study the specific types of locally convex spaces presently of interest to analysts, such as Frechet spaces, Q9,-spaces, Montel spaces, and nuclear spaces. It should be mentioned, however, that in almost all cases, the hypotheses of our general theorems are satisfied by these spaces. Aside from the overall approach, our main interest here is to study various types of Schauder bases which have previously been introduced, their relationships to each other and to various topological vector space concepts. In part I, we define the types of bases, interpret them in the context of a sequence space with a topology of uniform convergence on a family of subsets of the Kbthe dual and determine when a locally convex space can be viewed in this context. In part II we study the types of bases and their implications. The most important result is Theorem (1.10) which is a generalization of a theorem of R. C. James [9]. Finally in part III, we apply our results to the concepts of reflexivity and similar bases. Notation and terminology. If E[fT]; where E is a vector space over the field K of real or complex numbers and 9is a topology on E; is a locally convex space (always assumed to be Hausdorff) and (bn) is a sequence in E, we say that (bn) is a

Spectral decomposition of quasi-Montel spaces

In [8] the author showed that Montel spaces have the property that all regular Borel spectral measures with values in their continuous-linear-transformation algebras are necessarily purely atomic. The purpose of this note is to make the observation that by virtue of a theorem of Bartle, Dunford and Schwartz [1] and Grothendieck [3], this property is shared by a significantly larger class of locally convex spaces, namely the quasi-Montel spaces of K. Kera [5]. This class includes the classical Banach space 1 and its subspaces and the gestufte Raiume of Kothe and their subspaces [6]. The result given in this note also has consequences, which we shall mention briefly, in the study of the singular operators of Kantorovitz [4]. Let E [Z] be a locally convex topological vector space, which will be assumed to be boundedly complete; recall that a spectral measure triple in S(E) is a triple (X, 25, ,u) where X is a set, 25 is a a-algebra of subsets of X, and ,u: -->.C(E) is an S(E)-valued set function which is countably additive in the weak operator topology and for which ,u(X) = 1 (the identity transformation) and for any 5, E-C2, Mu(6'e) =-(8),u(E). (X, (25, ,u) is said to be equicontinuous if the values of ,u on (25 are, and to be Baire or Borel if X is a compact Hausdorff space and (5 = e or Q8, its a-algebras of Baire or Borel sets respectively. A Borel spectral measure triple (X, 03, ,u) is said to be regular if (Mu(.)x, x') is a regular Borel measure for each xCE and x'CE' (cf. [8, Proposition 3.18]). A point atom of a Borel measure is a point t CX with .( {I }) #0. For x CE, the cyclic subspace and real cyclic subspace generated by x, denoted by 91Q(x) and J)R(X) respectively, are the smallest closed subspace and closed real subspace of E respectively which contain { u(8)xl}es. [8, Proposition 3.15] shows that both 9NR(X) and 9(x) are complete locally convex spaces when E is boundedly complete in Z, [8, Proposition 3.13 et seq.] that )R(X) is a complete vector lattice when ordered by taking its positive cone to be the closed convex cone generated by { u(8)x1}sE, and also that 91Q(X) = =9RR(X) ED i fR(X) X It is easy to see that only small modifications of the proof of [8, Theorem 4.1] suffice to yield a proof of the slightly stronger-looking

Differential Equations and Differential Calculus in Montel Spaces

Differential Equations and Differential Calculus in Montel Spaces

Differential equations and differential calculus in Montel spaces

A topological vector space is a set which is equipped with a geometric structure and a topological structure. It is natural to ask if this is sufficient to derive an analytic structure, or if one has to impose it from without as in the case of manifolds. By an analytic structure, we mean the operations of differentiation and integration, and their corresponding properties. It is well established that integration can be defined reasonably (for a summary of the vast amount of literature on this subject, see Hyers [7]). In the case of differentiation, until recently, little has been done beyond the case of Banach spaces. In the latter case, the program has been completed by Hildebrandt and Graves [6], who developed the calculus, Kerner [8], who characterized the perfect differentials, Michal and Elconin [9], who extended the Picard existence and uniqueness theorems for differential equations, and Dieudonne [4], who showed that the weaker Peano existence theorem, which assumes only continuity, is false for Banach spaces. To begin the extension to non-normed spaces, Gil de Lamadrid [5], gave a definition of the derivative and started the development of the calculus. In the present paper, for the case of Montel spaces (to be defined below), we extend the calculus, characterize the perfect differentials and prove existence and uniqueness theorems for differential equations. In this paper, R will refer to the set of real numbers, 0 the empty set, and the symbol, 1, will indicate the completion of a proof. Iff and g are two functions with the property that the range of g is contained in the domain of f, then f o g will indicate the composition of these two functions. If A, B, are two sets, then will mean the set of all functions (in whatever function space is being considered) which map A into B. If C is another set, then o will be the set of all compositions,f og, withf ( and g E . It is obvious that o c .