Dunford-Pettis Property Research Articles

Gantmacher Type Theorems for Holomorphic Mappings

Given a holomorphic mapping of bounded type g ∈ Hb(U, F), where U ⊆ E is a balanced open subset, and E, F are complex Banach spaces, let A : Hb(F) ∈ Hb(U) be the homomorphism defined by A(f) = fog for all f ∈ Hb(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W ⊂ U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize the reflexivity of the space Hb(U,F) and prove that E' and F have the Schur property if and only if Hb(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.

Compactness of a Locally Compact Group G and Geometric Properties of Ap(G)

AbstractLet G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

On the Dunford-Pettis property in spaces of vector-valued bounded functions

We show that L∞(μ,X) has the Dunford-Pettis property for some classical Banach spaces including L1(μ), C (K), the disc algebra A and H∞.

𝐶(𝐾,𝐴) and 𝐶(𝐾,𝐻^{∞}) have the Dunford-Pettis property

Denote by X X either the disc algebra A A , or the space H ∞ H^{\infty } of bounded analytic functions on the disc, or any of their even duals. Then C ( K , X ) C(K,X) has the Dunford-Pettis property.

Banach space properties of $L\sp 1$ of a vector measure

We consider the space LI (v) of real functions which are integrable with respect to a measure v with values in a Banach space X. We study type and cotype for LI (v) . We study conditions on the measure v and the Banach space X that imply that LI (v) is a Hilbert space, or has the Dunford-Pettis property. We also consider weak convergence in L1 (v) .

Estimates by polynomials

Consider the following possible properties which a Banach space X may have: (P): If (xi) and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xj) − (yj) → 0, then Q(xj − yj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynomial Q on X.(RP): If (xj)and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xj − yj) → 0, then Q(xj) − Q(yj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynimial Q on X. We study properties (P) and (RP) and their relation with the Schur proqerty, Dunford-Pettis property, Λ, and others. Several applications of these properties are given.

Geometric Properties of Coefficient Function Spaces Determined by Unitary Representations of a Locally Compact Group

Let G be a locally compact group. As usual, let C*(G) denote the group C*-algebra of G and let B(G) denote its dual, the Fourier-Stieltjes algebra of G. If π is a (continuous) unitary representation of G, let Aπ be the space of coefficient functions of π and Bπ be its σ(B(G), C*(G))-completion in B(G). We first investigate the link between the Radon-Nikodým property for these coefficient function spaces and the complete reducibility of π. We then study the equivalence between the RNP and weak RNP for these spaces. In the next section, the Dunford-Pettis property for Bπ is characterised by the finite dimensionality of the irreducible representations weakly contained in π. The same characterisation holds for Aπ when VN(G), the group von Neumann algebra of G, has a nonzero type I, finite part. We note that the previous result cannot hold for all locally compact groups. We show finally that Aπ has the Schur property if and only if π is the direct sum of finite dimensional representations. At the end of each section, applications to the characterisation of certain classes of groups are given.

Banach space properties of strongly tight uniform algebras

We use the work of J. Bourgain to show that some uniform algebras of analytic functions have certain Banach space properties. If X is a Banach space, we say X is strong} if X and X* have the Dunford-Pettis property, X has the Pełczyński property, and X* i

The Dunford-Pettis Property for some Function Algebras in Several Complex Variables

The Dunford-Pettis Property for some Function Algebras in Several Complex Variables

When every polynomial is unconditionally converging

Letting $E$, $F$ be Banach spaces, the main two results of this paper are the following: (1) If every (linear bounded) operator $E\rightarrow F$ is unconditionally converging, then every polynomial from $E$ to $F$ is unconditionally converging (definition as in the linear case). (2) If $E$ has the Dunford-Pettis property and every operator $E\rightarrow F$ is weakly compact, then every $k$-linear mapping from $E^k$ into $F$ takes weak Cauchy sequences into norm convergent sequences. In particular, every polynomial from $\ell_\infty$ into a space containing no copy of $\ell_\infty$ is completely continuous. This solves a problem raised by the authors in a previous paper, where they showed that there exist nonweakly compact polynomials from $\ell_\infty$ into any nonreflexive space.

A Remark on the Dunford-Pettis Property in L 1 (μ, X)

We prove that if X is an Loo space, then LI (,u, X) has the Dunford-Pettis Property. In this note we shall devote our attention to the research of those Banach spaces having the Dunford-Pettis property [3] such that LI(jI, X) [3] verifies the same property. According to our knowledge, the results that are known so far are those due to Andrews [1] and Bourgain [2]. In particular, in Bourgain's paper it has been proved that LI(jI, C(K)) has the Dunford-Pettis property. Using this result, in the present note we show that if X is an Loo space [2], then LI(jI, X) has the Dunford-Pettis property too. First of all we prove the following Lemma. The space L1 (,u) 0, X** = L1 (j,u X**) is a closed subspace of the space (L1 (u) 07 X)** = L1 (A, X)**. Proof. It is easy to see that LI (,) , X** is a subset of (LI (,) 0, X)** and that 11 T'11(L1(,u)?&X)- 0, we can find a finite rank bounded linear operator v from LI(Cu) to L ICu) such that IIvH I 1 and IIv(fi) fill < e for all i=1, ..., n. Now we put E = v(L u)) and T, = Eni v(fi) 0xi** . Of course, Te E E o7r X** = E** o7r X**. Because E* is finite dimensional and the projective norm is accessible [4], we have E** 07r X** = Bc(E*, X*) = (E* o X*)*, where V is the injective norm. On the other hand, we have [4] (E o7r X)** = ((E o7r X)*)* = B(E, X)* = (E* o X*)*. Received by the editors October 7, 1991 and, in revised form, April 28, 1992. 1991 Mathematics Subject Classification. Primary 46E40, 46M05.

A remark on Dunford-Pettis property in 𝐿₁(𝜇,𝑋)

We prove that if X X is an L ∞ {L_\infty } space, then L 1 ( μ , X ) {L_1}(\mu ,X) has the Dunford-Pettis Property.

The Dunford-Pettis property on vector-valued continuous and bounded functions

Let X be a completely regular space, E a Banach space, Cb(X, E) the space of all continuous, bounded and E-valued functions defined on X, M(X, L(E, F)) the space of all L(E, F)-valued measures defined on the algebra generated by zero subsets of X. Weakly compact and β0-continuous operators defined from Cb(X, E) into a Banach space F are represented by integrals with respect to L(E, F)-valued measures. The strict Dunford-Pettis and the Dunford-Pettis properties are established on (Cb(X, E), βi), where βi denotes one of the strict topologies β0, β or β1, when E is a Schur space; the same properties are established on (Cb(X, E), β0), when E is an AM-space or an AL-space.

The Dunford-Pettis property is not a three-space property

An example of a Banach spaceX is shown which does not have the Dunford-Pettis property, while some subspaceY and the corresponding quotientX/Y have the hereditary Dunford-Pettis property.

WEAK SEQUENTIAL CONTINUITY AND THE DUNFORD-PETTIS PROPERTY

Abstract The paper gives a characterization of the Dunford-Pettis property of a Banach Space in terms of the joint weak sequential continuity of a composition mapping.

Weakly Continuous Mappings on Banach Spaces with the Dunford-Pettis Property

The Dunford-Pettis weak ( dw) topology on a Banach space is introduced as the finest topology that coincides with the weak topology on Dunford-Pettis sets. We characterize a wide class of polynomials between Banach spaces (including all the scalar valued polynomials) which are dw-continuous, and prove that a Banach space has the Dunford-Pettis (DP) property if and only if all these polynomials are weakly sequentially continuous. This result contains a characterization of the DP property obtained by Ryan, answering a question of Pelczyński: E has the DP property if and only if any weakly compact polynomial on E takes weak Cauchy sequences into convergent ones. It also extends other characterizations of the DP property by Operators to the case of polynomials. Similar results are given for holomorphic mappings. Other properties of polynomials and holomorphic mappings between Banach spaces are obtained.

Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems

Let G be a locally compact topological group and A(G) [B(G)] be, respectively, the Fourier and Fourier-Stieltjes algebras of G. It is one of the purposes of this paper to investigate the RNP (= Radon-Nikodym property) and some other geometric properties such as weak RNP, the Dunford-Pettis property and the Schur property on the algebras A(G) and B(G), and to relate these properties to the properties of the multiplication operator on the group C*-algebra C*(G). We also investigate the problem of Arens regularity of the projective tensor products C*(G)⊗A, when B(G) = C*(G)* has the RNP and A is any C*-algebra. Some related problems on the measure algebra, the group algebra and the algebras A p (G), PF p (G), PM p (G) (1 < p < ∞) are also discussed

On the tensor product of weakly compact operators

The identity operator on an infinite dimensional Hilbert space shows that neither the projective nor the injective tensor product of two weakly compact operators will be weakly compact in general; see Holub, [H1, Remark, p. 10]. There are, however, conditions ensuring this. Along the studies of Diestel and Faires, [DF], and Saksman and Tylli, [ST], we will add some more. Proposition 1, due to Saksman and Tylli, deals with the tensor product of a compact and a weakly compact operator. Answering a question of theirs, we consider in Proposition 2 the tensor product of two weakly compact operators on spaces with the Dunford-Pettis property. Having treated the case of absolutely p-summing operators in Proposition 3, we show in Proposition 4 that the injective tensor product of two weakly compact operators on Pisier spaces is weakly compact. Our results concern only the projective and the injective tensor product. For the elementary properties of weakly compact operators to be used here we refer to [DS, VI. 4, p. 482-5].

The Dunford-Pettis Property in the Predual of a Von Neumann Algebra

The von-Neumann algebras whose predual has the Dunford-Pettis property are characterised as being Type I finite. This answers the question asked by Chu and Iochum in The Dunford Pettis property in ${C^*}$-algebras, Studia Math. 97 (1990), 59-64.

Banach spaces in which Dunford-Pettis sets are relatively compact

Introduction. Let E be a Banach space and X a bounded subset of E. X is called a Dunford-Pettis set if for any weak null sequence (x*) c E* one has lim sup ix* (x)] = 0. n X This note is devoted to a study of the family of Banach spaces with the property that their Dunford-Pettis subsets are relatively compact; we shall say that such a space has the (DPrcP). Our interest in this class of Banach spaces is motivated by the following fact: in the paper [4] we proved that a dual Banach space with the Weak Radon-Nikodym Property ([8], in short (WRNP)) has the (DPrcP); since any Banach space with the (DPrcP) has the so called Compact Range Property ([8], in short (CRP)), it turns out that the result from [4] can be reversed, so obtaining a new characterization of the (WRNP) in dual spaces; this result makes the (WRNP) in dual spaces easier to be handled: for instance, we are able to answer a question by Ruess, [12], when defining a research program concerning projective tensor products of Banach spaces. There is another reason that could as well motivate our study: we state that dominated operators from special C(K, E) spaces taking values in a Banach space with the (DPrcP) are Dunford-Pettis; if E is finite dimensional it is well-known that this is always verified, but when the dimension of E is infinite the above result is no longer true. When looking for hypotheses on E and F making dominated operators Dunford-Pettis we realize that this happens if E has the Dunford-Pettis property ([2]) and F the (DPrcP); and these are in a sense the best hypotheses one can consider. All of these facts are contained in Section 1. Section 2 contains some more examples of Banach spaces with the (DPrcP) as well as some permanence results.