In this paper, our main goal is to define the class of weakly Demi Dunford-Pettis applications. We also study their relationship with the classes of weakly Dunford-Pettis and Demi Dunford-Pettis operators, including a condition where these operators coincide with Demi Dunford-Pettis and Demicompacts. In the last section, we study some of the behavior of this class in the Banach Lattice environment.
In this paper, we characterize Banach lattices on which each Dunford-Pettis operator (or weak Dunford-Pettis) is unbounded absolute weak Dunford-Pettis operator and the converse.
By introducing the concepts of order almost Dunford-Pettis and almost weakly limited operators in Banach lattices, we give some properties of them related to some well known classes of operators, such as, order weakly compact, order Dunford-Pettis, weak and almost Dunford-Pettis and weakly limited operators. Then, we characterize Banach lattices E and F on which each operator from E into F that is order almost Dunford-Pettis and weak almost Dunford-Pettis is an almost weakly limited operator.
In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated
We give a positive answer to the question of K. Bouras [`Almost Dunford-Pettis sets in Banach lattices', \textit{Rend. Circ. Mat. Palermo (2)} \textbf{ 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 $KB$-space.
We prove several results establishing conditions on the Banach lattices E_1,..., E_m and F so that the Aron-Berner extensions of (separately) almost Dunford-Pettis m-linear operators from E_1 x ... x E_m to F are (separately) almost Dunford-Pettis. Illustrative examples are provided.
Let $A$ be the disk algebra, $Ω$ be a compact Hausdorff space and $μ$ be a finite Borel measure in $Ω$. It is shown that the dual of $C(Ω,A)$ has the Dunford-Pettis Property. This proved in particular that the spaces $C(Ω,A)$ and $L^1(μ,A*)$ have the Dunford-Pettis property.
We introduce and study the enveloping norms of regularly P-operators, where P is an "almost" version of limited, Grothendieck, and of Dunford--Pettis operators in Banach lattices. Several further topics related to these operators are also discussed.
In the present paper we study unconditionally $p$-converging operators and Dunford-Pettis property of order $p$. New characterizations of unconditionally $p$-converging operators and Dunford-Pettis property of order $p$ are established. Six quantities are defined to measure how far an operator is from being unconditionally $p$-converging. We prove quantitative versions of relationships of completely continuous operators,unconditionally $p$-converging operators and unconditionally converging operators. We further investigate possible quantifications of the Dunford-Pettis property of order $p$.
We investigate possible quantifications of the Dunford-Pettis property. We show, in particular, that the Dunford-Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford-Pettis property. We prove that $L^1$ spaces and $C(K)$ spaces posses both of them. We also show that several natural measures of weak non-compactness are equal in $L^1$ spaces.
We prove in particular that Banach spaces of the form $C_0(Ω)$, where $Ω$ is a locally compact space, enjoy a quantitative version of the reciprocal Dunford-Pettis property.
We characterize those compact Hausdorff spaces K such that the proyective tensor product of C(K) by itself has the Dunford-Pettis Property, answering thus in the negative a question posed by Castillo and Gonzalez.
We introduce a class of operators which is called unbounded Dunford-Pettis. In this paper, we study some properties of the operators, relationships with the other classes of operators and the space of these operators.
In this article, the class of all Dunford-Pettis $ p $-convergent operators and $ p $-Dunford-Pettis relatively compact property on Banach spaces are investigated. Moreover, we give some conditions on Banach spaces $ X $ and $ Y $ such that the class of bounded linear operators from $ X$ to $ Y $ and some its subspaces have the $ p $-Dunford-Pettis relatively compact property. In addition, if $ Ω$ is a compact Hausdorff space, then we prove that dominated operators from the space of all continuous functions from $ K $ to Banach space $ X $ (in short $ C(Ω,X) $) taking values in a Banach space …
We introduce and study the class of unbounded DunfordPettis operators. As consequences, we give basic properties and derive interesting results about the duality, domination problem and relationship with other known classes of operators.
The $p$-Gelfand Phillips property ($1\le p<\infty$) is studied in spaces of operators. Dunford - Pettis type like sets are studied in Banach spaces. We discuss Banach spaces $X$ with the property that every $p$-convergent operator $T:X\to Y$ is weakly compact, for every Banach space $Y$.
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 adjoint 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. The stability of reciprocal Dunford-Pettis property of order \(p\) for the projective tensor product is given.
A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the $P_N$ Dunford--Pettis property if and only if every weakly compact $N-$homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric c…
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: (i) $X$ is a hemicompact space, (ii) $X$ is a cosmic space (…
We prove that if $X,Y$ are Banach spaces, $Ω$ a compact Hausdorff space and $U\hbox{\rm :} C(Ω,X)\to Y$ is a bounded linear operator, and if $U$ is a Dunford--Pettis operator the range of the representing measure $G(Σ) \subseteq DP(X,Y)$ is an uniformly Dunford--Pettis family of operators and $\|G\|$ is continuous at $\emptyset$. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space $C([0,1],X)$ with values in $c_{0}$ or $l_{p}$, ($1\leq p<\infty$) be Dunford--Pettis and/or compact operators, in which, Khinchin's inequalit…
We show that the dual to any subspace of $c_0(Γ)$ has the strongest possible quantitative version of the Schur property. Further, we establish relationship between the quantitative Schur property and quantitative versions of the Dunford-Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on $\ell_p$ ($1<p<\infty$) with Dunford-Pettis property satisfies automatically both its quantitative versions.
In this paper, using the concept of unbounded absolute weak convergence ($uaw$-convergence, for short) in a Banach lattice, we define two classes of continuous operators, named $uaw$-Dunford-Pettis and $uaw$-compact operators. We investigate some properties and relations between them. In particular, we consider some hypotheses on domain or range spaces of operators such that the adjoint or the modulus of a $uaw$-Dunford-Pettis or $uaw$-compact operator inherits a similar property. In addition, we look into some connections between compact operators, weakly compact operators, and Dunford-Pettis…
In this article, we introduce the notion of $p$-$(DPL)$ sets.\ Also, a factorization result for differentiable mappings through Dunford-Pettis $p$-convergent operators is investigated.\ Namely, if $ X ,Y $ are real Banach spaces and $U$ is an open convex subset of $X,$ then we obtain that, given a differentiable mapping $f: U\rightarrow Y$ its derivative $f^{\prime}$ takes $U$-bounded sets into $p$-$(DPL)$ sets if and only if it happens $f=g\circ S,$ where $S$ is a Dunford-Pettis $p$-convergent operator from $X$ into a suitable Banach space $Z$ and $g:S(U)\rightarrow Y$ is a Gâteaux differenti…
Using the hierarchy of weakly null sequences introduced by Argyros, Merkourakis, and Tsarpalias, we introduce two new families of operator classes. The first family simultaneously generalizes the completely continuous operators and the weak Banach-Saks operators. The second family generalizes the class $\mathfrak{DP}$. We study the distinctness of these classes, and prove that each class is an operator ideal. We also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. We produce a number of examples, including the higher …
We characterize the points of $\left\|\cdot\right\|$-$w^*$ continuity of dual maps, turning out to be the smooth points. We prove that a Banach space has the Schur property if and only if it has the Dunford-Pettis property and there exists a dual map that is sequentially $w$-$w$ continuous at $0$. As consequence, we show the existence of smooth Banach spaces on which the dual map is not $w$-$w$ continuous at $0$.
We prove some general results on sequential convergence in Fréchet 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-Pet…
We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that $(c_0\hat{\otimes}_πc_0)^{**}$ fails the DPP. Since $(c_0\hat{\otimes}_πc_0)^{*}$ does enjoy it, this provides a new space with the DPP whose dual fails to have it. We also prove that, if $E$ and $F$ are ${\mathscr L}_1$-spaces, then $E\hat{\otimes}_εF$ has the DPP if and only if both $E$ and $F$ have the Schur property. Other results and examples are given.
Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent spaces of bounded operators on $X$. In this paper, we consider some situations under which these classes of bounded operators form locally solid vector lattices. In addition, we generalize some notions of $uaw$-Dunford-Pettis operators and $uaw$-compact operators defined on a Banach lattice to general theme of locally solid vector lattices. With the aid of appropriate topologies, we investigate some relations between topological and lattice structures of these operators. In particular, we character…
Recently, J. H'michane et al. introduced the class of weak* Dunford-Pettis operators on Banach spaces, that is, operators which send weakly compact sets onto limited sets. In this paper the domination problem for weak* Dunford-Pettis operators is considered. Let $S, T:E\rightarrow F$ be two positive operators between Banach lattices $E$ and $F$ such that $0\leq S\leq T$. We show that if $T$ is a weak$^{*}$ Dunford-Pettis operator and $F$ is $σ$-Dedekind complete, then $S$ itself is weak* Dunford-Pettis.
We give biorthogonal system characterizations of Banach spaces that fail the Dunford-Pettis property, contain an isomorphic copy of $c_0$, or fail the hereditary Dunford-Pettis property. We combine this with previous results to show that each infinite dimensional Banach space has one of three types of biorthogonal systems.
In this paper, we first study the concept of $ p $-sequentially Right property, which is the $ p$-version of the sequentially Right property. Also, we introduce a new class of subsets of Banach spaces which is called $ p$-Right$ ^{\ast} $ set and obtain the relationship between p-Right subsets and p-Right$ ^{\ast} $ subsets of dual spaces. Furthermore, for $ 1\leq p<q\leq\infty, $ we introduce the concepts of properties $ (SR)_{p,q}$ and $ (SR^{\ast})_{p,q}$ in order to find a condition which every Dunford-Pettis $ q $-convergent operator is Dunford-Pettis $p$-convergent. Finally, we apply the…
We introduce a weakened version of the Dunford-Pettis property, and give examples of Banach spaces with this property. In particular, we show that every closed subspace of Schreier's space $S$ enjoys it. As an application, we characterize the weak polynomial convergence of sequences, show that every closed subspace of $S$ has the polynomial Dunford-Pettis property of Biström et al. and give other polynomial properties of $S$.
We prove that the class of Banach function lattices in which all relatively weakly compact sets are equi-integrable sets (i.e. spaces satisfying the Dunford-Pettis criterion) coincides with the class of 1-disjointly homogeneous Banach lattices. A new examples of such spaces are provided. Furthermore, it is shown that Dunford-Pettis criterion is equivalent to de la Vallee Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.
In this paper, we study $un$-dual (in symbol, $\ud{E}$) of Banach lattice $E$ and compare it with topological dual $E^*$. If $E^*$ has order continuous norm, then $E^* = \ud{E}$. We introduce and study weakly unbounded norm topology ($wun$-topology) on Banach lattices and compare it with weak topology and $uaw$-topology. In the final, we introduce and study $wun$-Dunford-Pettis opertors from a Banach lattice $E$ into Banach space $X$ and we investigate some of its properties and its relationships with others known operators.
In this paper, we present some necessary and sufficient conditions for semi-compact operators being almost L-weakly compact (resp. almost M-weakly compact) and the converse. Mainly, we prove that if $X$ is a nonzero Banach space, then every semi-compact operator $T: X\rightarrow E$ is almost L-weakly compact if and only if the norm of $E$ is order continuous. And every positive semi-compact operator $T:E\rightarrow F$ is almost M-weakly compact if and only if the norm of $E'$ is order continuous. Moreover, we investigate the relationships between almost L-weakly compact operators and Dunford-P…
We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak$^{*}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and $L$-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into $c_{0}$, we give an operator characterization of those $σ$-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a $σ$-Dedekind Banach lattice $E$, every relatively…
In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spac…
In this paper, we will study some properties of b-weakly compact operators and we will investigate their relationships to some variety of operators on the normed vector lattices. With some new conditions, we show that the modulus of an operator $T$ from Banach lattice $E$ into Dedekind complete Banach lattice $F$ exists and is $b$-weakly operator whenever $T$ is a $b$-weakly compact operator. We show that every Dunford-Pettis operator from a Banach lattice $E$ into a Banach space $X$ is b-weakly compact, and the converse holds whenever $E$ is an $AM$-space or the norm of $E^\prime$ is order co…
Let $F$ be a function with values in a Banach space. When $F$ is locally (Pettis or Bochner) integrable with respect to a locally determined positive measure, a vector measure $ν_F$ with density $F$ defined on a $δ$-ring is obtained. We present the existing connection between the spaces $L^1_w(ν_F)$, $L^1(ν_F)$ and $L^1(|ν_F|)$ and the spaces of Dunford, Pettis or Bochner integrable functions.
We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices $E$, $F$ for which every positive weak almost limited operator $T:E\rightarrow F$ is almost limited (resp. almost Dunford-Pettis). As consequences, we will give some interesting results.
In this paper we introduce and study a new class of operators related to norm bounded sets on Banach Lattice and which brings together several classical classes of operators (as o-weakly compact operators, b-weakly compact operators, M-weakly compact operators, L-weakly compact operators, almost Dunford-Pettis operators). As consequences, we give some new lattice approximation properties of these classes of operators.
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 $L_p(μ)$-spaces 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.
Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of $c_{0}$ is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford-Pettis property are $\mathbfΠ^{1}_{2}$-complete.
