Spaces Of Vector-valued Continuous Functions Research Articles

Compact Multiplication Operators on Weighted Spaces of Vector-Valued Continuous Functions

In this note we characterize the compact multiplication operators Mπ on the weighted locally convex spaces CV0(X,E) of vector-valued continuous functions induced by the operator-valued mappings π : X → B(E). 0. Introduction. Let L(X,E) be a vector space of functions from a nonempty set X to a vector space E over the field C or R. Let T (E) be a set of linear transformations from E to itself. If φ : X → X and π : X → T (E) are mappings such that the weighted composite function π.f ◦ φ belongs to L(X,E), whenever f ∈ L(X,E), then the mapping taking f to π.f ◦φ is a linear transformation on L(X,E) and we denote it by Wπ,φ. In case L(X,E) is a topological vector space and the mapping Wπ,φ is continuous, we call Wπ,φ the weighted composition operator on L(X,E) induced by the symbol (π, φ). In case φ is the identity map, we call it the multiplication operator induced by π and we denote it by Mπ. For details on these operators we refer to [11]. The compact weighted composition operators on spaces of continuous functions have been studied extensively by many authors like Kamowitz [5], Feldman [3], Singh and Summers [13], Jamison and Rajagopalan [4], Takagi [14], Chan [2] and Singh and Manhas [12]. As we know, the class of weighted composition operators include the class of multiplication operators and the class of composition operators. One natural question arises: is it possible to get the behavior of the compact multiplication operators from the study of the compact weighted composition operators? In general, it is not possible since the conditions obtained earlier for a weighted composition operator to be compact is not satisfied by the identity map φ. So it motivates us to 1991 AMS Mathematics Subject Classification. Primary 47B38, 47B07, Secondary 47A56, 46E10, 46E40.

Polynomially significant properties and equivalence of topologies on fully nuclear spaces

We relate the equivalence of the topologies τ o and τ ω on a fully nuclear space E having the bounded approximation property with the polynomially significant properties on E′ b , using the localization property of Defant and Govaerts (A. Defant and W. Govaerts (1986). Tensor products and spaces of vector-valued continuous functions. Manuscripta Math., 55, 433–449). This allows us to give examples of Fréchet nuclear spaces with bases E and F so that τ o =τ ω on . We also give an example of Fréchet nuclear spaces with bases E and F so that τ o =τ ω on for every open polydisc U in E× F′ ] . The conditions for equivalence of topologies are expressed in terms of the linear invariants (DN), and given in Vogt (D. Vogt (1983). Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist. J. reine u. angew. Math., 345, 182–200.) and Meise and Vogt (R. Meise and D. Vogt (1986). Holomorphic functions of uniformly bounded type on nuclear Fréchet spaces. Studia Math., 83, 147–175.).

Realcompactness and Banach-Stone theorems

For realcompact spaces $X$ and $Y$ we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on $X$ and $Y$ in two cases: the spaces of all continuous functions and the spaces of {\em bounded} continuous functions. With similar techniques we also describe the linear biseparating maps defined between some other families of spaces, in particular spaces of vector-valued uniformly continuous bounded functions.

On CL-spaces and almost CL-spaces

We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by c0- and l1-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

On CL-spaces and almost CL-spaces

We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by c0- and l1-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

Automatic continuity of biseparating maps

We prove that a biseparating map between spaces of vector-valued continuous functions is usually automatically continuous. However, we also discuss special cases when this is not true.

Realcompactness and spaces of vector-valued functions

It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions $A(X,E)$ and $A(Y,F)$ implies that some compactifications of $X$ and $Y$ are homeomorphic. In some cases, conditions are given to wa

DP1 and completely continuous operators

W. Freedman introduced an alternate to the Dunford‐Pettis property, called the DP1 property, in 1997. He showed that for 1 ≤ p < ∞, has the DP1 property if and only if each Xα does. This is not the case for . In fact, we show that has the DP1 property if and only if it has the Dunford‐Pettis property. A similar result also holds for vector‐valued continuous function spaces.

Weighted composition operators on nonlocally convex weighted spaces of continuous functions

LetV be a system of weights on a completely regular Hausdorff spaceX and letB(E) be the topological vector space of all continuous linear operators on a general topological vector spaceE. LetCV0(X, E) andCVb(X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operatorsWπ,ϕ onCV0(X, E) (orCVb(X, E)) induced by the operator-valued mappings π:X→B(E) (or the vector-valued mappings π:X→E, whereE is a topological algebra) and the self-map ϕ ofX. Also, we characterize the mappings π:X→B(E) (or π:x→E) and ϕ:X→X which induce the compact weighted composition operators on these weighted spaces of continuous functions.

Weak Sequential Convergence and Weak Compactness in Spaces of Vector-Valued Continuous Functions

Let X be completely regular Hausdorff space, E a Hausdorff locally convex space, C(X, E) (Cb(X, E)) the space of all E-valued (all E-valued bounded) continuous functions on X, and βz a strict topology on Cb(X, E). It is proved that a sequence {ƒn} in (Cb(X, E), βz) converges weakly to 0 if and only if {ƒn} is uniformly bounded and ƒn,(x) → 0, pointwise on X with weak topology on E. Similar results are proved for (C(X, E), β∞c). Some characterizations of relatively weakly compact subsets of (Cb(X, E), βz) and (C(X, E), β∞c) are also given.

Bornological spaces of null sequences

It is proved that c0(E) is an LB-space, whenever E is either a Montel LB-space or a coechelon space k∞(v) of type ∞ (a partial solution of a problem of Schmets and Bierstedt). In the second case we show that c0(k∞(v)) is even the Mackey completion of c0 ⊗e k∞(v). Introduction. J. Schmets has successfully studied various linear-topological properties of spaces of vector-valued continuous functions [Sch2]. Unfortunately, the problem of bornologicity of such spaces has been left open (comp. [Sch1]). Up to now a solution has been obtained only in very special cases (see [Bo1], [Mu], [Sch2, I.7.3 and IV.4.7], [DG], [BoS1] and [BoS2]), in particular, it is still unclear when c0(E) is bornological. Bierstedt and Schmets asked if c0(E), E a DFM-space, is bornological (or, equivalently, an LB-space). We solve this problem completely. Then we show that the problem of bornologicity of c0(E) for general LB-spaces E (also asked by Schmets and Bierstedt [Sch1], see also [Sch2, Ch. IV]) reduces in some sense to the same problem for Montel E. By this reduction we solve the problem for arbitrary coechelon spaces E = k∞(v). We use our “factorization” approach developed in [DiDo1] and [DiDo2]. The problem of boronologicity of C(K,E), K compact, at least in case when E = indn∈N En is an LBor LF-space, is closely related to the question of interchanging of e-products and inductive limits (comp. [BoS2]). Indeed, by a result of Mujica ([Mu], [Sch2, I.7.2]), it is known that C(K,E) = C(K)e indn∈N En contains a bornological space indn∈N(C(K)eEn) as a topological subspace. Thus, if (∗) C(K)e ind n∈N En = ind n∈N (C(K)eEn) holds algebraically, then it holds topologically and C(K,E) is a fortiori bornological (see the result of Mujica [Sch2, I.7.3] or Marquina and Schmets [MS] and a more general result of Defant and Govaerts [Sch2, IV.4.7], [DG, Th. 13, Cor. 14]). If the space E is compactly regular (i.e., each compact subset of E is compact in some step En), then (∗) holds. Roughly speaking, for LB-spaces the case of 1991 Mathematics Subject Classification. Primary 46A11, 46A13, 46E40.

Multiplication Operators and Dynamical Systems

AbstractLet X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.

P-adic Ascoli theorems and compactoid polynomials

We prove several properties of compactoid sets in some spaces of vector-valued continuous functions. As a consequence we derive some p-adic Ascoli theorems which allow us to answer the question proposed by W.H. Schikhof in [MR.90i 46128].

Multiplication operators on weighted spaces of vector-valued continuous functions

AbstractIf V is a system of weights on a completely regular Hausdorff space X and E is alocally convex space, then CV0(X, E) and CVb (X, E) are locally convex spaces of vector-valued continuous functions with topologies generated by seminorms which are weighted analogues of the supremum norm. In this paper we characterise multiplication operators on these spaces induced by scalar-valued and vector-valued mappings. Many examples are presented to illustrate the theory.

On weak approximation and convexification in weighted spaces of vector-valued continuous functions

Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the formwhere υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.

Operators on locally convex spaces of vector-valued continuous functions

LetEandFbe locally convex spaces and letKbe a compact Hausdorff space.C(K,E) is the space of allE-valued continuous functions defined onK, endowed with the uniform topology.Starting from the well-known fact that every linear continuous operatorTfromC(K,E) toFcan be represented by an integral with respect to an operator-valued measure, we study, in this paper, some relationships between these operators and the properties of their representing measures. We give special treatment to the unconditionally converging operators.As a consequence we characterise the spacesEfor which an operatorTdefined onC(K,E) is unconditionally converging if and only if (Tfn) tends to zero for every bounded and converging pointwise to zero sequence (fn) inC(K,E).

Tensor products and spaces of vector-valued continuous functions

Quasibarrelled, barrelled and bornological tensor products of locally convex spaces are studied. A device, called the desintegration lemma, is developed for the most difficult case, that of the injective topology. Applications are given to spaces of vector-valued continuous functions.

Characterization of some classes of operators on spaces of vector-valued continuous functions

LetKbe a compact Hausdorff space andE, FBanach spaces. We denote byC(K, E)the Banach space of all continuous.E-valued functions defined onK,with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator)TfromC(K, E)toFhas a finitely additive representing measuremof bounded semi-variation, defined on the Borel σ-field Σ ofKand with values inL(E, F″)(the space of all operators fromEinto the second dual ofF),in such a way thatwhere the integral is considered in Dinculeanu's sense.

Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions

Given a compact Hausdorff spaceX, EandFtwo Banach spaces, letT: C(X, E) → Fdenote a bounded linear operator (hereC(X, E)stands for the Banach space of all continuousE-valued functions defined onXunder supremum norm). It is well known [4] that any such operatorThas a finitely additive representing measureGthat is defined on the σ–field of Borel subsets ofXand thatGtakes its values in the space of all bounded linear operators fromEinto the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance[1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that ifE*andE**have the Radon-Nikodym property then a bounded linear operatorT: C(X, E) → Fis weakly compact if and only if the measureGis continuous at Ø (also called strongly bounded), i.e. limn||G|| (Bn) = 0 for every decreasing sequenceBn↘ Ø of Borel subsets ofX(here ||G|| (B) denotes the semivariation ofGatB), and if for every Borel setBthe operatorG(B)is a weakly compact operator fromEtoF. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties thenE*andE**must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators onC(X, E). It is known [6] that ifT: C(X, E) → Fis an unconditionally converging operator, then its representing measureGis continuous at 0 and, for every Borel setB, G(B)is an unconditionally converging operator fromEtoF. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spacesEwith the Schur property in terms of properties of Dunford-Pettis operators onC(X, E)spaces.

A barrelledness criterion forC 0(E)

Denote by C0(E) the linear space of all sequences in E which converge to zero in E endowed with its natural topology. A. Marquina and J. M. Sanz Serna [same journal 31 (1978/79), 589–596; MR0531574 (80i:46010)] gave necessary and sufficient conditions to ensure the quasibarrelledness of C0(E), namely: (i) E is quasibarrelled and (ii) the strong dual of E satisfies condition (B) of Pietsch. Moreover, if E is complete in the sense of Mackey, (i) and (ii) characterize the barrelledness of C0(E). Even in the absence of completeness in the sense of Mackey, the last result is true as the article under review shows by a clever use of a "sliding-hump'' technique. Interesting consequences of those results in the theory of tensor products and spaces of vector-valued continuous functions can be found in a recent paper by A. Defant and W. Govaerts ["Tensor products and spaces of vector-valued continuous functions'', Manuscripta Math., to appear] and in the monograph of J. Schmets [Spaces of vector-valued continuous functions, Lecture Notes in Math., 1003, Springer, Berlin, 1983].