For each $S \in L(E)$ (with $E$ a Banach space) the operator $R(S) \in L(E^{**}/E)$ is defined by $R(S)(x^{**}+E) = S^{**}x^{**}+E$ \quad ($x^{**}\in E^{**}$). We study mapping properties of the correspondence $S\to R(S),$ which provides a representation $R$ of the weak Calkin algebra $L(E)/W(E)$ (here $W(E)$ denotes the weakly compact operators on $E$). Our results display strongly varying behaviour of $R.$ For instance, there are no non--zero compact operators in Im$(R)$ in the case of $L^1$ and $C(0,1),$ but $R(L(E)/W(E))$ identifies isometrically with the class of lattice regular operators on $\ell^2$ for $E=\ell^2(J)$ (here $J$ is the James' space). Accordingly, there is an operator $T \in L(\ell^2(J))$ such that $R(T)$ is invertible but $T$ fails to be invertible modulo $W(\ell^2(J)).$
We introduce the class of unbounded $M$-weakly operators and the class of unbounded $L$-weakly compact operators. We investigate some properties for these new classification of operators and we study relation between them and $M$-weakly compact and $L$-weakly compact operators. We also present an operator characterization of Banach lattices with order continuous norm. \keywords{unbounded $M$-weakly compact \and unbounded $L$-weakly compact \and unbounded norm convergence \and $M$-weakly compact \and $L$-weakly compact
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…
Based on the concept of unbounded absolutely weakly convergence, we give new characterizations of L-weakly compact sets. As applications, we find some properties of order weakly compact operators. Also, a new characterizations of order continuous Banach lattices are obtained.
In this paper, we introduce and study new concepts of order L-weakly and order M-weakly compact operators. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have order continuous norms. It is proved that if $T : E \longrightarrow F$ is an operator between two Banach lattices, then T is order M-weakly compact if and only if its adjoint T' is order L-weakly compact. Also, we show that if its adjoint T' is order M-weakly compact, then T is order L-weakly compact. Some related results are also obtained.
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.
Several recent papers investigated lattice copies and unbounded convergences in Banach lattices. In this paper, we first solve the problem of RV and LA which is an extension of the well-known James distortion theorem. Using lattice copies and unbounded convergence, then we introduce weak L- and M-weakly compact operators on Banach lattices and research the relationship between these operators and L- and M-weakly compact operators. Finally, we study the compactness of weak L-weakly compact and weak M-weakly compact operators.
A locally compact group $ G $ is discrete if and only if the Fourier algebra $ A(G) $ has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let $H$ be an ultraspherical hypergroup and let $A(H)$ denote the corresponding Fourier algebra. We will give several characterizations of discreteness of $ H $ in the terms of the algebraic properties of $A(H)$. We also study Arens regularity of closed ideals of $ A(H)$.
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…
A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show that the positive operator $T$ from $E$ into Dedekind complete $F$ is almost order-weakly compact if and only if $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of operators and its relationships with others known operators.
Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat{f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behavior of weakly convergent sequences made of finitely supported elements in Lipschitz-free sp…
For $0\leqslant ξ\leqslant ω_1$, we define the notion of $ξ$-weakly precompact and $ξ$-weakly compact sets in Banach spaces and prove that a set is $ξ$-weakly precompact if and only if its weak closure is $ξ$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant ξ\leqslant ω_1$, we prove that a Banach space $X$ has the $ξ$-Schur property if and only if every $ξ$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.
We construct an Asplund Banach space $\mathcal{X}$ with a norming Markuševič basis such that $\mathcal{X}$ is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wiesław Kubiś and Arkady Leiderman.
In the paper Henstock, McShane, Birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general Banach space X. In particular the existence of selections integrable in the same sense of the corresponding multifunctions has been considered.
The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the closed convex hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of spe…
Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that $\operatorname{LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact convex subset of a Banach space. We show that there always exists a common fixed point for any jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$. Several variances involving the weak* compactness, the RNP, the distality of $K$ and/or the left reversibility of $S$ are also provided.
A. Bahramnezhad and K. Haghnejad Azar introduced the classes of $KB$-operators and $WKB$-operators, and they studied some of theirs properties. In the present paper, we give answer for an open problem from that paper, which two classifications of operators, $b$-weakly compact operators and $KB$-operators are different. A continuous operator $T$ from a normed vectoe lattice $E$ into a normed space $X$ is said to be $KB$-operator (respectively, $WKB$-operator) if $\{Tx_n\}_n$ has a norm (respectively, weak) convergent subsequence in $X$ for every positive increasing sequence $\{x_n\}_n$ in the c…
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.
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 $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spac…
In this paper, using the concept of unbounded absolute weak convergence ($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…
Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_α\xrightarrow{Fo}0$ implies $Tx_α\xrightarrow{\Vert.\Vert}0$ for each $(x_α)_α\subseteq E$. Our mean from the convergence $ x_α\stackrel{Fo} {\longrightarrow} x $ is that there exists another net $ \left(y_α\right) $ in $F $ with the same index set satisfying $ y_α\downarrow 0 $ in $F$ and $ \vert x_α- x \vert \leq y_α$ for all indexes $ α$. In this paper, we will study some properties of this new class o…
We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such a topology is strictly finer than the coarse wedge topology and it turns every chain complete, rooted tree into a Fréchet--Urysohn, countably compact topological space. We show the rôle of such topology in the theory of weakly Corson and weakly Valdivia compacta. In particular, we give the first example of a compact space $T$ whose every closed subspace is weakly Valdivia, yet $T$ is not weakly Corson. This answers a question due to Ondřej Kalenda.
Several recent papers investigated unbounded convergences in Banach lattices. Combine all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. Some related results are obtained as well.
In this paper, we show that the sum of a compact convex subset and a simultaneously $τ$-strongly proximinal convex subset (resp. simultaneously approximatively $τ$-compact convex subset) of a Banach space X is simultaneously tau-strongly proximinal (resp. simultaneously approximatively $τ$-compact ), and the sum of weakly compact convex subset and a simultaneously approximatively weakly compact convex subset of X is still simultaneously approximatively weakly compact, where $τ$ is the norm or the weak topology. Moreover, some related results on the sum of simultaneously proximinal subspaces ar…
