Dunford Pettis Sets Research Articles

Some results on the class of weakly sequentially precompact sets and operators

The paper contains some results on weakly sequentially precompact sets and operators. In particular, we establish some relationships between weakly sequentially precompat operators and those whose the adjoint map (L) sets into relatively norm compact ones. Besides, we characterize the class of weak* Dunford-Pettis operators through weakly sequentially precompact operators and deduce in the sequel a new characterization of Dunford-Pettis* property. Moreover, we generalize [9, Theorem 2.5.9] and show that order weakly compact operators carry almost order Dunford-Pettis sets into weakly sequentially precompact ones. Furthermore, we prove that the product of order weakly compact operators and b-weakly compact ones maps weakly sequentially precompact sets into relatively weakly compact ones. Finally, we present some results about the positive Schur property.

The strong Dunford-Pettis relatively compact property of order p

We introduce and study Banach lattices with the strong Dunford-Pettis relatively compact property of order p (1 ? p < ?); that is, spaces in which every weakly p-compact and almost Dunford-Pettis set is relatively compact. We also introduce the notion of the weak Dunford-Pettis property of order p and then characterize this property in terms of sequences. In particular, in terms of disjoint weakly compact operators into c0, an operator characterization of those Banach lattices with the weak Dunford-Pettis property of order p is given. Moreover, some results about Banach lattices with the positive Dunford-Pettis relatively compact property of order p are presented

English

The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between LDunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces.

Polynomial versions of almost Dunford-Pettis sets and almost limited sets in Banach lattices

Let A be a norm bounded solid subset of a Banach lattice E and n be any positive integer. We prove that A is an almost Dunford-Pettis set if and only if every positive weakly compact n-homogeneous polynomial from E to c0 maps A to a relatively compact set in c0. Moreover, if E is σ-Dedekind complete, we also prove that A is an almost limited set if and only if every positive n-homogeneous polynomial from E to c0 maps A to a relatively compact set in c0.

Dunford–Pettis Sets, $$V^*$$ V ∗ -Sets, and Property $$(MB^*)$$ ( M B ∗ )

In this paper, we study how some localized properties, e.g., Dunford–Pettis sets, and $$V^*$$ -sets can be used to study more global structure properties. We introduce a new class of Banach spaces called Banach spaces with property $$(MB^*)$$ . We say that a space X has property $$(MB^*)$$ if every $$V^*$$ -set in X is a Dunford–Pettis set in X. We characterize those spaces which have property $$(MB^*)$$ . It is shown that X has property $$(MB^*)$$ if and only if for every Banach space Y, every unconditionally converging adjoint operator $$T^*$$ from $$X^*$$ to $$Y^*$$ is completely continuous. Also, we recall property (MB) for Banach spaces and then study relation between these two properties.

A note on weak reciprocal Dunford–Pettis sets

A bounded subset K of X is defined to be a weak reciprocal Dunford–Pettis set (or wRDP set) if T(K) is relatively weakly compact for each completely continuous operator \({T \colon X \to c_0}\). A Banach space X has the R* property if every wRDP subset of X is relatively weakly compact. In this paper we study weak reciprocal Dunford–Pettis sets and Banach spaces with property R*.

ON A QUESTION OF BOURAS CONCERNING WEAK COMPACTNESS OF ALMOST DUNFORD–PETTIS SETS

We give a positive answer to the question of Bouras [‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227–236] concerning weak compactness of almost Dunford–Pettis sets in Banach lattices. That is, every almost Dunford–Pettis set in a Banach lattice $E$ is relatively weakly compact if and only if $E$ is a $\mathit{KB}$-space.

Almost Dunford–Pettis sets in Banach lattices

We introduce and study the class of almost Dunford–Pettis sets in Banach lattices. It also discusses some of the consequences derived from this study. As an application, we characterize Banach lattices whose relatively weakly compact sets are almost Dunford–Pettis sets. Also, we establish some necessary and sufficient conditions on which an almost Dunford–Pettis set is L-weakly compact (respectively, relatively weakly compact). In particular, we characterize Banach lattices under which almost Dunford–Pettis sets in the topological dual of a Banach lattice coincide with that of L-weakly compact (respectively, relatively weakly compact) sets. As a consequences we derive some results.

Dunford–Pettis Properties and Spaces of Operators

AbstractJ. Elton used an application of Ramsey theory to show that ifXis an infinite dimensional Banach space, thenc0embeds inX, ℓ1embeds inX, or there is a subspace ofXthat fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that ifc0embeds inX*, then ℓ∞embeds inX*. Emmanuele and John showed that ifc0embeds inK(X,Y), thenK(X,Y) is not complemented inL(X,Y). Classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The spaceLw*(X*,Y) ofw*–wcontinuous operators is also studied.

Errata and addendum: tensor products and Dunford–Pettis sets

AbstractGhenciu and Lewis introduced the notion of a strong Dunford–Pettis set and used this notion to study the presence or absence of isomorphic copies of c0 in Banach spaces. The authors asserted that they could obtain a fundamental result of J. Elton without resorting to Ramsey theory. While the stated theorems are correct, unfortunately there is a flaw in the proof of the first theorem in the paper which also affects subsequent corollaries and theorems. The difficulty is discussed, and Elton's results are employed to establish a Schauder basis proposition which leads to a quick proof of the theorem in question. Additional results where questions arise are discussed on an individual basis.

Strong Dunford-Pettis sets and spaces of operators (Monatsh. Math. 144, 275–284 (2005))

In a recent paper, Ghenciu and Lewis studied strong Dunford-Pettis sets and made the following two assertions: (1) The Banach space X* contains a nonrelatively compact strong Dunford-Pettis set if and only if l∞ ↪ X*. (2) If c0 ↪ Y and H is a complemented subspace of X so that H* is a strong Dunford-Pettis space, then W(X, Y) is not complemented in L(X, Y). While the statements are correct, the proofs are flawed. The difficulty with the proofs is discussed, and a fundamental result of Elton is used to establish a simple lemma which leads to quick proofs of both (1) and (2).

Tensor products and Dunford–Pettis sets

The structure of Dunford–Pettis sets, strong Dunford–Pettis sets and certain spaces of operators is studied in the context of the injective and projective tensor products of Banach spaces.

Strong Dunford?Pettis Sets and Spaces of Operators

Bibasic sequences of Singer are used to show that l1 embeds complementably in the Banach space X if and only if X* contains a non-relatively compact strong Dunford–Pettis set. Spaces of operators and strongly additive vector measures are also discussed.

Dunford–Pettis sets

Bibasic sequences are used to study relative weak compactness and relative norm compactness of Dunford-Pettis sets. A Banach space X has the Dunford-Pettis property provided that every weakly compact operator with domain X and range an arbitrary Banach space Y maps weakly compact sets in X into norm compact sets in Y. Localizing this notion, a bounded subset A of X is said to be a Dunford-Pettis subset of X if T(A) is relatively norm compact in Y whenever T: X -Y is a weakly compact operator. Consequently, a Banach space X has the Dunford-Pettis property if and only if each of its weakly compact sets is a Dunford-Pettis set. The survey article by Diestel [5] is an excellent source of information about classical results in Banach spaces which relate to the Dunford-Pettis property. Kevin Andrews utilized Dunford-Pettis sets in a study of the Bochner integral in [1]. In Theorem 1 of [1], Andrews showed that a subset K of X is a Dunford-Pettis subset of X if and only if limn (sup{ jx* (x) I x C K}) = 0 whenever (x*) is a weakly null sequence in X* ( = the continuous linear dual of X). In Corollary 4 of [1], Andrews used this characterization to show that if X has the Dunford-Pettis property and ?1 does not embed in X, then the space Ll(pLi,X) has the Dunford-Pettis property. E. Bator showed in [2] that a dual space has the weak Radon-Nikodym property if and only if each Dunford-Pettis subset of X is relatively compact. In addition to reproducing Bator's result, Emmanuele [8] established several other structure properties for Banach spaces in which all Dunford-Pettis sets are relatively compact. Since every bounded subset of a Banach space X whose dual space X* has the Schur property is a Dunford-Pettis subset of X, it is clear that there are Dunford-Pettis sets which are not relatively weakly compact. However, we note that Odell [13, p. 377] showed that every sequence in a Dunford-Pettis set has a weakly Cauchy subsequence. In this paper we study Dunford-Pettis sets which fail to be relatively norm or weakly compact. The following definitions and notation will be helpful. A sequence (Xnv, f2) in XxX* is called bibasic [14, p. 85], [4] if (xn) is a basic sequence in X, (fn) is a basic sequence in X*, and fi*(xj) = ?ij. If (Xn, fn) is a bibasic sequence, Xo = [Xn], Received by the editors April 14, 1998 and, in revised form, March 15, 2000. 2000 Mathematics Subject Classification. Primary 46B20; Secondary 46B15, 46B45.

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.

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.

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.