Properties Of Banach Spaces Research Articles

Unbounded Banach-Saks Operators and Unbounded Grothendieck Operators on Banach Lattices

Recently, there has been much attention to the ordered structures beyond Banach lattices. Moreover, we have many nice properties in Banach spaces that can be transformed naturally into Banach lattice cases. Therefore, combining these notions with order structure can produce nicer notions, as well. Suppose $E$ is a Banach lattice. Recently, there have been some motivating contexts regarding the known Banach--Saks property and the Grothendieck property from an order point of view. In this paper, we establish these results for operators that enjoy different types considered for the Banach-Saks property. We characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding operator versions related to the Banach--Saks properties. Moreover, we consider different notions related to the Grothendieck property from an ordered attitude; then, we investigate operator versions of these concepts, as well. In particular, beside other results, we characterize order continuity and reflexivity of the underlying Banach lattices in terms of the corresponding bounded linear operators defined on the corresponding Banach lattices, as well.

Asymptotic smoothness, concentration properties in Banach spaces and applications

We prove an optimal result of stability under ℓp-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.

On norm derivatives and the ball-covering property of Banach spaces

We study a local version of the ball-covering problem in Banach spaces, and obtain a complete solution to it in terms of the norm derivatives. We illustrate the advantage of the local approach by obtaining substantial refinements of several previously known results on this topic.

On the p-Dunford–Pettis relatively compact property of Banach spaces

On the p-Dunford–Pettis relatively compact property of Banach spaces

On the property (C) of Corson and other sequential properties of Banach spaces

A well-known result of R. Pol states that a Banach space X has property (C) of Corson if and only if every point in the weak*-closure of any convex set C⊆BX⁎ is actually in the weak*-closure of a countable subset of C. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of BX⁎ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under MA+¬CH for the class of C(K)-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom (PFA) the following conditions are equivalent for an arbitrary Banach space X:(1)X has property E′;(2)X has weak*-sequential dual ball;(3)X has property (C) of Corson;(4)(BX⁎,w⁎) has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property E′ has weak*-convex block compact dual ball.

Approximation properties in terms of Lipschitz maps

We investigate some approximation properties of Banach spaces which are described in terms of Lipschitz maps. First, we present characterizations of the Lipschitz approximation property, and prove that a Banach space $X$ has the approximation property whe

The Dunkl-Williams constant related to the singer orthogonality and red isosceles orthogonality in Banach spaces

In this paper, we shall consider two new constants DWS(X) and DWI(X), which are the Dunkl-Williams constant related to the Singer orthogonality and theisosceles orthogonality, respectively. We discuss the relationships between DWS(X) and some geometric properties of Banach spaces, including uniform non-squareness, uniform convexity. Furthermore, an equivalent form of DWS(X) in the symmetric Minkowski planes is given and used to compute the value of DWS((R2, ???p)), 1 < p < ?, and we also give a characterization of the Radon plane with affine regular hexagonal unit sphere in terms of DWS(X). Finally, we establish some estimates for DWI(X) and show that DWI(X) does not necessarily coincide with DWS(X).

Asymptotic smoothness in Banach spaces, three-space properties and applications

We study four asymptotic smoothness properties of Banach spaces, denoted T p , A p , N p \mathsf {T}_p,\mathsf {A}_p, \mathsf {N}_p , and P p \mathsf {P}_p . We complete their description by proving the missing renorming characterization for A p \mathsf {A}_p . We show that asymptotic uniform flattenability (property T ∞ \mathsf {T}_\infty ) and summable Szlenk index (property A ∞ \mathsf {A}_\infty ) are three-space properties. Combined with the positive results of the first-named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three-space problem for this family of properties. We also derive from our characterizations of A p \mathsf {A}_p and N p \mathsf {N}_p in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes.

Quantifying properties (K) and (μs$\mu ^{s}$)

AbstractA Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence so that as for every weakly null sequence in X; X has property if every weak* null sequence in admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.

Some properties of $I_3$-λ-statistical cluster points

In this note, we investigate some problems concerning the set of I_3I 3–\lambdaλ-statistical cluster points of triple sequences via ideals in finite dimensional spaces, and some of its properties in finite dimensional Banach spaces are proved.

Some Results on the $p$-Weak Approximation Property in Banach Spaces

In this study, some existing results dealing with the weak approximation property of Banach spaces are considered for the $p$-weak approximation property. Also, an observation on the bounded weak approximation and the $p$-bounded weak approximation properties is given. Moreover, the proof of the solution of the duality problem for the $p$-weak approximation property which exists in the literature is given in a shorter way as an alternative.

The relations between the von Neumann–Jordan type constant and some geometric properties of Banach spaces

The relations between the von Neumann–Jordan type constant and some geometric properties of Banach spaces

Alternative proofs of some classical metric fixed point theorems by using approximate fixed point sequences

The notion of approximate fixed point sequence, emphasized in Chidume (Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer-Verlag London, Ltd., London, 2009), is a very useful tool in proving convergence theorems for fixed point iterative schemes in the class of nonexpansive-type mappings. In the present paper, our aim is to present simple and unified alternative proofs of some classical fixed point theorems emerging from Banach contraction principle, by using a technique based on the concepts of approximate fixed point sequence and graphic contraction.

Geometry of Banach Spaces: A New Route Towards Position Based Cryptography

In this work we initiate the study of position based quantum cryptography (PBQC) from the perspective of geometric functional analysis and its connections with quantum games. The main question we are interested in asks for the optimal amount of entanglement that a coalition of attackers have to share in order to compromise the security of any PBQC protocol. Known upper bounds for that quantity are exponential in the size of the quantum systems manipulated in the honest implementation of the protocol. However, known lower bounds are only linear. In order to deepen the understanding of this question, here we propose a position verification (PV) protocol and find lower bounds on the resources needed to break it. The main idea behind the proof of these bounds is the understanding of cheating strategies as vector valued assignments on the Boolean hypercube. Then, the bounds follow from the understanding of some geometric properties of particular Banach spaces, their type constants. Under some regularity assumptions on the former assignment, these bounds lead to exponential lower bounds on the quantum resources employed, clarifying the question in this restricted case. Known attacks indeed satisfy the assumption we make, although we do not know how universal this feature is. Furthermore, we show that the understanding of the type properties of some more involved Banach spaces would allow to drop out the assumptions and lead to unconditional lower bounds on the resources used to attack our protocol. Unfortunately, we were not able to estimate the relevant type constant. Despite that, we conjecture an upper bound for this quantity and show some evidence supporting it. A positive solution of the conjecture would lead to stronger security guarantees for the proposed PV protocol providing a better understanding of the question asked above.

Lipschitz retractions and complementation properties of Banach spaces

In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space F(X) is a Plichko space whenever X is a Plichko Banach space. Our main results include two examples of metric spaces. The first one M contains two points {0,1} such that no separable subset of M containing these points is a Lipschitz retract of M. The second example fails the analogous property for arbitrary infinite density. Finally, we introduce the metric version of the concept of locally complemented Banach subspace, and prove some metric analogues to the linear theory.

Limited $p$-converging operators and relation with some geometric properties of Banach spaces

By using the concepts of limited $p$-converging operators between two Banach spaces $X$ and $Y$, $L_p$-sets and $L_p$-limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$-Dunford--Pettis property of order $p$ and Pelczyński's property of order $p$, $1\leq p<\infty$.

Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions

We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. We subsequently use this geometric characterization to answer several questions in Lipschitz analysis. Notably, it follows that the Lipschitz-free spaceF(M)\mathcal {F}(M)over a compact metric spaceMMis a dual space if and only ifMMis purely 1-unrectifiable. Furthermore, we establish a compact determinacy principle for the Radon-Nikodým property (RNP) and deduce that, for any complete metric spaceMM, pure 1-unrectifiability is actually equivalent to some well-known Banach space properties ofF(M)\mathcal {F}(M)such as the RNP and the Schur property. A direct consequence is that any complete, purely 1-unrectifiable metric space isometrically embeds into a Banach space with the RNP. Finally, we provide a possible solution to a problem of Whitney by finding a rectifiability-based description of 1-critical compact metric spaces, and we use this description to prove the following: a bounded turning tree fails to be 1-critical if and only if each of its subarcs hasσ\sigma-finite Hausdorff 1-measure.

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 &lt; ∞). 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

A Godefroy—Kalton principle for free Banach lattices

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded linear right-inverse must have a lattice homomorphism right-inverse. In terms of free Banach lattices, this can be rephrased into the following question: which Banach lattices embed into the free Banach lattice which they generate as a lattice-complemented sublattice? We will provide necessary conditions for a Banach lattice to have the lattice-lifting property, and show that this property is shared by Banach spaces with a 1-unconditional basis as well as free Banach lattices. The case of C(K) spaces will also be analyzed.

Small combination of slices, dentability and stability results of small diameter properties in Banach spaces

In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in [9], [4], [12], [13] was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in [14]. We introduce the Ball Huskable Property (BHP), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, BSCSP namely, the unit ball has convex combination of slices of arbitrarily small diameter and BDP namely, the closed unit ball has slices of arbitrarily small diameter. We show BDP implies BHP which in turn implies BSCSP and none of the implications can be reversed. We prove similar results for the w⁎-versions. We prove that all these properties are stable under lp sum for 1≤p≤∞,c0 sum and Lebesgue Bochner spaces. Finally, we explore the stability of these with properties in the light of three space property. We show that BHP is a three space property provided X/Y is finite dimensional and same is true for BSCSP when X has BSCSP and X/Y is strongly regular ([14]).