Dunford-Pettis Property Research Articles

Some results on the class of U-Dunford-Pettis operators

In this paper, we show that for a pair of Banach lattices E and F with E Dedekind σ-complete, the following statements are equivalent:(1)Each positive U-Dunford-Pettis (respectively, Dunford-Pettis) operator T:E→F is AM-compact;(2)Either the norm of E is order continuous or else F is discrete and the norm of F is order continuous.If, in addition, F has the weak Dunford-Pettis property, then each U-Dunford-Pettis (respectively, Dunford-Pettis) operator T:E→F is AM-compact if and only if either the norm of E is order continuous or else F has the Schur property. As a consequence, we obtain some new characterizations of Banach lattices with order continuous norms (respectively, with the Schur property).

Locally Convex Spaces with Sequential Dunford–Pettis Type Properties

Let p,q,q′∈[1,∞], q′≤q. Several new characterizations of locally convex spaces with the sequential Dunford–Pettis property of order (p,q) are given. We introduce and thoroughly study the sequential Dunford–Pettis* property of order (p,q) of locally convex spaces (in the realm of Banach spaces, the sequential DP(p,∞)* property coincides with the well-known DPp* property). Being motivated by the coarse p-DP* property and the p-Dunford–Pettis relatively compact property for Banach spaces, we define and study the coarse sequential DP(p,q)* property, the coarse DPp* property and the p-Dunford–Pettis sequentially compact property of order (q′,q) in the class of all locally convex spaces.

Disjoint Dunford–Pettis-Type Properties in Banach Lattices

ABSTRACT New characterizations of the disjoint Dunford–Pettis property of order p (disjoint DPPp) are proved and applied to show that a Banach lattice of cotype p has the disjoint DPPp whenever its dual has this property. The disjoint Dunford–Pettis$^*$ property of order p (disjoint $DP^*P_p$) is thoroughly investigated. Close connections with the positive Schur property of order p, with the disjoint DPPp, with the p-weak $DP^*$ property and with the positive $DP^*$ property of order p are established. In a final section, we study the polynomial versions of the disjoint DPPp and of the disjoint $DP^*P_p$.

ON SOME PROPERTIES OF BANACH SPACE-VALUED FIBONOCCI SEQUENCE SPACES

In this work, we give some results about the basic properties of the vector-valued Fibanocci sequence spaces. In general, sequence spaces with Banach space-valued cannot have a Schauder Basis unless the terms of the sequences are complex or real terms. Instead, we de ned the concept of relative basis in [11] by generalizing the de nition of a basis in Banach spaces. Using this de nition, we can characterize certain important properties of vector- term Fibanocci sequence spaces, such as separability, Dunford-Pettis Property, approximation property, Radon-Riesz Property and Hahn-Banach extension property.

On Some Sequence Spaces via q-Pascal Matrix and Its Geometric Properties

We develop some new sequence spaces 𝓁p(P(q)) and 𝓁∞(P(q)) by using q-Pascal matrix P(q). We discuss some topological properties of the newly defined spaces, obtain the Schauder basis for the space 𝓁p(P(q)) and determine the Alpha-(α-), Beta-(β-) and Gamma-(γ-) duals of the newly defined spaces. We characterize a certain class (𝓁p(P(q)),X) of infinite matrices, where X∈{𝓁∞,c,c0}. Furthermore, utilizing the proposed results, we characterize certain other classes of infinite matrices. We also examine some geometric properties, like the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and Banach–Saks-type p property of the spaces 𝓁p(P(q)) and 𝓁∞(P(q)).

Ergodicity of commuting multioperators and holomorphic multioperators of multiplication

In this paper, the strong ergodic theorems are extended from the case of one bounded operator to the case of commuting multioperators. The authors show that in Grothendieck space with the Dunford-Pettis property, mean ergodic operator, and uniform ergodic operator are the same. We study when multioperators of multiplication on a weighted Banach space of holomorphic multi-functions are power bounded, mean ergodic, or uniformly mean ergodic.

Some characterizations of almost p-convergent operators and applications

In the present paper, the notion of almost p-(V) sets is defined to characterize order bounded almost p-convergent operators from a Banach lattice E into another Banach lattice F. Also, for 1≤p≤q≤∞ we introduce the concept of almost q-(V) property of order p on Banach lattices in order to find an operator theoretic characterization of q-weakly compactness in terms of almost p-(V) sets. Moreover, some characterizations of those Banach lattices with the disjoint Dunford-Pettis property of order p are investigated. Finally, we introduce the class of almost weak p-convergent operators and we give a characterization of these operators in terms of weakly compact operators and the adjoint of almost p-convergent operators.

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

Topological properties of some multiplication operators on L(X)

A pair (u, ?) in X ? X?, where X is an infinite dimensional Banach space and X? its topological dual space, induces in a natural way two multiplication operators L?,u and R?,u on the Banach space L(X), defined by L?,u (T) (x) = ? (T(x)) u, and R?,u (T) (x) = ?(x)T(u), for all T in L(X) and x in X. In this paper, we present necessary and sufficient conditions for the compactness, demicompactness, stongly demicompactess, power compactness and Riesz property of this family of operators. We also establish sufficient conditions for the quasi-compactness and weak compactness of these operators. Finally, we show that the Dunford-Pettis property fails for the Banach space L(X) whenever either X or L(X) is reflexive.

(Non-)Dunford–Pettis operators on noncommutative symmetric spaces

We fully characterize the class of von Neumann algebras M (on a separable Hilbert space) such that any non-Dunford–Pettis operator on the predual M⁎ fixes a copy of ⊕ℓ1(ℓ2), which extends results due to Bourgain [9] and Rosenthal [49]. We also fully characterize those noncommutative symmetric spaces affiliated with different types of von Neumann algebras, which have the Dunford–Pettis property. This extends earlier results for the special cases (e.g., for symmetric function spaces, and von Neumann algebras and their preduals), by Bunce [11], by Chu and Iochum [15], and by Kaminska and Mastylo [36]. Finally, we supply new examples of symmetric spaces with the DPP on general (non-resonant) σ-finite measure spaces.

WMB-PROPERTY OF ORDER p IN BANACH SPACES

In this paper, we introduce a new property of Banach spaces called wMB-property of order p (1 ≤ p < ∞). A necessary and sufficient condition for a Banach space to have the wMB-property of order p is given. We study p-convergent operators and weakly-p-L-sets. Banach spaces with the wMB-property of order p are characterized. Also, the Dunford-Pettis property of order p and DP∗ -property of order p are studied in Banach spaces. Finally we show the relation between Pelczynski’s property (V ) and wMB-property of order p

Cosine families in GDP Quojection-Fréchet spaces

We prove that if the Quojection-Fréchet space $X$ is a Grothendieck space with the Dunford-Pettis property, then every $C_ {0}$-cosine family is necessarily uniformly continuous and therefore its infinitesimal generator is a continuous linear operator.

Polynomial versions of weak Dunford–Pettis properties in Banach lattices

Polynomial versions of weak Dunford–Pettis properties in Banach lattices

The positive polynomial Schur property in Banach lattices

We study the class of Banach lattices that are positively polynomially Schur. Plenty of examples and counterexamples are provided, lattice properties of this class are proved, arbitrary L p ( μ ) L_p(\mu ) -spaces, 1 ≤ p > ∞ 1 \leq p > \infty , are shown to be positively polynomially Schur, lattice analogues of results on Banach spaces are obtained and relationships with the positive Schur and the weak Dunford-Pettis properties are established.

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

We prove some general results on sequential convergence in Frechet lattices that yield, as particular instances, the following results regarding a closed ideal \(I\) of a Banach lattice \(E\): (i) If two of the lattices \(E\), \(I\) and \(E/I\) have the positive Schur property (the Schur property, respectively) then the third lattice has the positive Schur property (the Schur property, respectively) as well; (ii) If \(I\) and \(E/I\) have the dual positive Schur property, then \(E\) also has this property; (iii) If \(I\) has the weak Dunford-Pettis property and \(E/I\) has the positive Schur property, then \(E\) has the weak Dunford-Pettis property. Examples and applications are provided.

The Pełczyński and Dunford–Pettis properties of the space of uniform convergent Fourier series with respect to orthogonal polynomials

The Pełczyński and Dunford–Pettis properties of the space of uniform convergent Fourier series with respect to orthogonal polynomials

V -sets and the property ( V L D ) in Banach spaces

In this paper, we study the notion of V-sets in Banach spaces and Banach lattices, and we give some characterizations of it in terms of sequences. As an application, we establish new properties of unconditionally converging operators and 1-Schur property in Banach lattices. Next, by introducing the concept of the property $(VLD)$ in Banach spaces, we investigate the Dunford-Pettis completely continuous property of unconditionally converging operator. Finally, we derive the relationships between the property $(VLD)$ and the relatively compact Dunford-Pettis property (resp., the Pelczynski's property $(V)$), and we deduce some examples of Banach spaces with the property $(VLD)$. <br><br> У цій роботі ми вивчаємо поняття V-множин у банахових просторах та банахових ґратках та даємо деякі їхні характеристики у термінах послідовностей. Як застосування, ми встановлюємо нові властивості безумовно збіжних операторів і властивість 1-Шура в банахових ґратках. Далі, вводячи поняття $(VLD)$ властивості у банахових просторах, ми досліджуємо властивість цілковитої неперервності Данфорда-Петтіса безумовно збіжного оператора. Нарешті, ми виводимо зв’язки між $(VLD)$ властивістю і властивістю відносної компактності Данфорда-Петтіса (відповідно, властивістю Пельчинського $(V)$), і виводимо деякі приклади банахових просторів із $(VLD)$ властивістю.

Weak compactness of almost L-weakly and almost M-weakly compact operators

In this paper, we investigate conditions on a pair of Banach lattices E and F that tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operator T : E → F is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator T : E → F is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space X is almost L-weakly compact, then E is a KB-space or X has the Dunford-Pettis property and the norm of E is order continuous.

A study of reciprocal Dunford–Pettis-like properties on Banach spaces

In this article, we study the relationship between p- (V) subsets and p-$$ (V^{*})$$ subsets of dual spaces. We investigate the Banach space X with the property that the adjoint of every p-convergent operator $$ T:X\rightarrow Y $$ is weakly q-compact, for every Banach space Y. Moreover, we define the notion of q-reciprocal Dunford–Pettis$$ ^{*} $$ property of order p on Banach spaces and obtain a characterization of Banach spaces with this property. Also, the stability of reciprocal Dunford–Pettis property of order p for projective tensor product is given.

Dunford–Pettis type properties and the Grothendieck property for function spaces

For a Tychonoff space X, let $$C_k(X)$$ and $$C_p(X)$$ be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that $$C_k(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property. We extend Grothendieck’s result by showing that $$C_k(X)$$ has both the Dunford–Pettis property and the sequential Dunford–Pettis property if X satisfies one of the following conditions: (1) X is a hemicompact space, (2) X is a cosmic space (= a continuous image of a separable metrizable space), (3) X is the ordinal space $$[0,\kappa )$$ for some ordinal $$\kappa $$ , or (4) X is a locally compact paracompact space. We show that if X is a cosmic space, then $$C_k(X)$$ has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that $$C_p(X)$$ has the Dunford–Pettis property and the sequential Dunford–Pettis property for every Tychonoff space X, and $$C_p(X) $$ has the Grothendieck property if and only if every functionally bounded subset of X is finite.