Peter G. Casazza, Niels Jorgen Nielsen
In this paper we investigate a Gaussian average property of Banach spaces. This property is weaker than the Gordon Lewis property but closely related to this and other unconditional structures. It is also shown that this property implies that certain Hilbert space valued operators defined on subspaces of the given space can be extended.
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In this paper we first take a detail survey of the study of the Banach-Saks property of Banach spaces and then show the Banach-Saks property of the product spaces generated by a finite number of Banach spaces having the Banach-Saks property. A more general inequality for integrals of a class of composite functions is also given by using this property.
A Banach space E is said to have Property (w) if every (bounded linear) operator from E into E' is weakly compact. We give some interesting examples of James type Banach spaces with Property (w). We also consider the passing of Property (w) from E to C(K,E).
Uniformly nonsquare Banach spaces have the fixed point property 1
本文提出了一个新的证明，论证了一致非方Banach空间具有不动点性质。
We establish the weak Banach-Saks property for function spaces arising as the optimal domain of an operator.
A Banach space X is said to have the Tsirelson property if it does not contain subspaces that are isomorphic to l_{p}, p in [1,infty) or c_{0}. The article contains a quite simple method to producing Banach spaces with the Tsirelson property.
The aim of this note is to consider different notions for the Banach-Saks property in locally solid vector lattices as an extension for the known concepts of the Banach-Saks property in Banach lattices. We investigate relations between them; in particular, we shall characterize spaces in which, these notions agree.
More on Property($M$) in separable Banach spaces: Revised
本文探讨了可分Banach空间中可与Property($M$)及Property($M^*$)相关联的Banach空间性质，深化了度量不动点理论的研究。
Uniformly nonsquare Banach spaces have the Fixed Point Property 2
本文为一致非方 Banach 空间具有不动点性质这一结论提供了一个新的证明。
Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence ( $uaw$-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded oper…
These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounde…
We present a generalization of the Radon-Riesz property to sequences of continuous functions with values in uniformly convex and uniformly smooth Banach spaces.
Reconstruction property in Banach spaces introduced and studied by Casazza and Christensen in [1]. In this paper we introduce reconstruction property in Banach spaces which satisfy $\mathfrak{I}$-property. A characterization of reconstruction property in Banach spaces which satisfy $\mathfrak{I}$-property in terms of frames in Banach spaces is obtained. Banach frames associated with reconstruction property are discussed.
We extend the well-known criterion of Lotz for the dual Radon-Nikodym property (RNP) of Banach lattices to finitely generated Banach $C(K)$-modules and Banach $C(K)$-modules of finite multiplicity. Namely, we prove that if $X$ is a Banach space from one of these classes then its Banach dual $X^\star$ has the RNP iff $X$ does not contain a closed subspace isomorphic to $\ell^1$.
We consider a Banach space, which comes naturally from c0 and it appears in the literature, and we prove that this space has the fixed point property for non-expansive mappings.
Uniformly nonsquare Banach spaces have the Fixed Point Property
本文针对一致非方 Banach 空间的不动点性质，提出了一种新的证明方法，为该经典结论提供了替代论证。
We present many examples of Banach spaces of linear operators and homogeneous polynomials without the approximation property, thus improving results of Dineen and Mujica [11] and Godefroy and Saphar [13].
We give a characterisation of the separable Banach spaces with the Daugavet property which is applied to study the Daugavet property in the projective tensor product of an $L$-embedded space with another non-zero Banach space. The former characterisation also motivates the introduction of two indices related to the Daugavet property and a short study of them.
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 as well as for different notions related to the Grothendieck property. In particular, beside other results, we characterize order continuity and reflexivity of Banach lattices in terms of the corresponding bounded operators defined on them.
We are interested in a sufficient condition given in an article by P. Lefèvre, É. Matheron and A. Primot to obtain the Blum-Hanson Property and we then partially answer two questions asked in this same article on other possible conditions to have this property for a separable Banach space.
The strong Nakano property in Banach lattices
该研究聚焦Banach格的强Nakano性质，证明有限维Banach空间对应的自由Banach格具该性质且常数与维数无关，还给出自由Banach格具该性质时原空间的子空间约束条件。
The weak Banach-Saks property for Hölder spaces
本文研究了Hölder空间的弱Banach-Saks性质，证明了在无限度量空间上该性质不成立。
Let $A$ and $B$ be commutative semisimple Banach algebras. In this paper, the $BSE$ and $BED$- property of tensor Banach algebra $A{\otimes}_γ B$ with respect to the Banach algebras $A$ and $B$ are assesed. In particular, $BSE$ and $BED$- structure of vector-valued group algebra $L_{1}(G, A)$ and $C^{*}$- algebra $C_{0}(X, A)$ characterized.
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
The main result of this paper establishes that the known Arazy-Cwikel property holds for classes of uniformly K-monotone spaces in the quasi-Banach setting provided that the initial couple is mutually closed. As a consequence, we get that the class of all quasi-Banach K-spaces (i.e., interpolation spaces which are described by the real K-method) with respect to an arbitrary mutually closed Banach couple enjoys the Arazy-Cwikel property. Another consequence complements some previous results by Bykov and Ovchinnikov, showing that this property holds also for the class of all interpolation quasi-…
The principal aim of this paper is to study the Banach-Saks property when the speed of convergence of a Cesaro mean sequence can be chosen independtly from the choice of the initial sequence. We establish links between the uniform Banach-Saks property, properties $(A_\infty)$ of Partington and the $p$-Banach-Saks property. We also prove that the $p$-Banach-Saks property and the strong $p$-Banach-Saks property are equivalent. Many examples are given and we apply the results to the symmetric Kottman constant.
We prove that a Banach lattice $X$ which does not contain the $l^n_{\infty}$-uniformly has an equivalent norm which is uniformly Kadec-Klee for a natural topology $τ$ on $X$. In case the Banach lattice is purely atomic, the topology $τ$ is the coordinatewise convergence topology.
We continue the investigation of the behaviour of diameter two properties in tensor products of Banach spaces. Our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products. We also prove a result for the symmetric strong diameter two property for the injective tensor product.
In this work, we introduce the notion of Ball dentable property in Banach spaces. We study certain stability results for the $w^*$-Ball dentable property leading to a discussion on Ball dentability in the context of ideals of Banach spaces. We prove that the $w^*$-Ball-dentable property can be lifted from an $M$-ideal to the whole Banach Space. We also prove similar results for strict ideals of a Banach space. We note that the space $C(K,X)^*$ has $w*$-Ball dentable property when $K$ is dispersed and $X^\ast$ has the $w^*$-Ball dentable property.
We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and $\ell_1$-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for $\ell_1$-spreading models.
It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
On Hahn-Banach smoothness and related properties in Banach spaces
本文研究Banach空间中Hahn-Banach光滑性的三类变体，刻画了相关性质，推导了强弱性质等价的充分条件，证明其可分决定性，并刻画了经典序列空间中对应子空间的特征。
We show that a minor refinement of the Bourgain-Rosenthal construction of a Banach space without the Radon-Nikodym property which contains no bounded $δ$-trees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such spaces; in particular we show that the Daugavet property is not inherited by ultraproducts.
The study of the Banach-Saks property in Banach spaces has a long and illustrious history. Of late, motivated by applications in financial mathematics, interest has arisen in the Banach-Saks type properties with respect to order convergence. This paper presents a study of order Banach-Saks properties in Banach function spaces, and in particular in rearrangement invariant spaces. Among the results obtained, we provide some sufficient conditions for the (weak) order Banach-Saks property. We also characterize the (weak) order Banach-Saks property in Orlicz spaces. It is also shown that the (weak)…
It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable space. Furthermore, every separable reflexive Banach space is a quotient of a reflexive complementably $\ell_p$ saturated space with $1<p<\infty$ and of a $c_0$ saturated space.
We demonstrate the result stated in the title, thus answering an open question asked by Julio Becerra Guerrero, Ginés López-Pérez and Abraham Rueda Zoca in J. Conv. Anal. \textbf{25}, no. 3 (2018).
We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again this property.
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-c…
Uniform Embeddings of Banach Space Spheres into Hilbert Spaces and Property (H)
本文证明可分无穷维实 Banach 空间单位球面均匀嵌入 Hilbert 空间蕴含该空间具有 Kasparov–Yu 性质 (H)。
We give an intrinsic characterisation of the separable reflexive Banach spaces that embed into separable reflexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht.
The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then $B(\ell_{\infty},X^*)=Π_2(\ell_{\infty},X^*)$. If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could be a Banach lattice) or if X is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension property for these classe…
In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $Δ$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $Γ$ with $\vert Γ\vert \geq 2$, the Banach space $\bigoplus_{γ\in Γ}^{c_0} X_γ$ satisfies the Mazur--Ulam property, whenever the Banach space $X_γ$ is strictly convex with dim$((X_γ)_{\mathbb{R}})\geq 2$ for every $γ$. Moreover we prove that the Banach space $C_0(K,X)$ satisfi…
In this paper we first show that if $X$ is a Banach space and $α$ is a left invariant crossnorm on $\ell_\infty\otimes X$, then there is a Banach lattice $L$ and an isometric embedding $J$ of $X$ into $L$, so that $I\otimes J$ becomes an isometry of $\ell_\infty\otimes_αX$ onto $\ell_\infty\otimes_m J(X)$. Here $I$ denotes the identity operator on $\ell_\infty$ and $\ell_\infty\otimes_m J(X)$ the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to characterize the Gordon-Lewis property $\GL$ in terms of emb…
In this note the following is proved. Separable L-embedded spaces - that is separable Banach spaces which are complemented in their biduals such that the norm between the two complementary subspaces is additive - have property (X) which, by a result of Godefroy and Talagrand, entails uniqueness of the space as a predual.
We present an overview to the approximation property, paying especial attention to the recent results relating the approximation property to ideals of linear operators and Lipschitz ideals. We complete the paper with some new results on approximation of Lipschitz mappings and their relation to linear operator ideals.
It is known that any separable Banach space with BAP is a complemented subspace of a Banach space with a basis. We show that every operator with bounded approximation property, acting from a separable Banach space, can be factored through a Banach space with a basis.
In the present paper we prove that a necessary condition for a Banach space $X$ to admit a generating compact Lipschitz retract $K$, which satisfies an additional mild assumption on its shape, is that $X$ enjoys the Bounded Approximation Property. This is a partial solution to a question raised by Godefroy and Ozawa.
In this paper, we provide a sublinear function $p$ on ordered Banach spaces, which depends on the order structure of the space. With respect to this $p$, we study the relation between $p$-contractivity of positive semigroups and the $p$-dissipativity of its generators. The positive off-diagonal property of generators is also studied in ordered vector spaces.
We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric $π$-property. Besid…
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.
A. Szankowski's example is used to construct a Banach space similar to that of "An example of an asymptotically Hilbertian space which fails the approximation property", P.G. Casazza, C.L. Garc\'ıa, W.B. Johnson [math.FA/0006134 (math@arXiv.org)].
We prove that the following three properties for a Banach space are all different from each other: every finite convex combination of slices of the unit ball is (1) relatively weakly open, (2) has nonempty interior in relative weak topology of the unit ball, and (3) intersects the unit sphere. In particular, the $1$-sum of two Banach spaces does not have property (1), but it has property (2) if both the spaces have property (1); the Banach space $C[0,1]$ does not have property (2), although it has property (3).
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.
Given a finite measure space $(Ω,Σ,μ)$, we show that any Banach space $X(μ)$ consisting of (equivalence classes of) real measurable functions defined on $Ω$ such that $f χ_A \in X(μ) $ and $ \|f χ_A \| \leq \|f\|, \, f \in X(μ), \ A \in Σ$, and having the subsequence property, is in fact an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.
On norm derivatives and the ball-covering property of Banach spaces
本文研究 Banach 空间球覆盖问题的局部形式，利用范数导数给出完整刻画，并据此改进多项已有结果。
We give a characterization of the existence of copies of $c_{0}$ in Banach spaces in terms of indexes. As an application, we deduce new proofs of James Distortion theorem and Bessaga-Pełczynski theorem about weakly unconditionally Cauchy series.
W.B. Johnson has constructed a series of Banach spaces non isomorphic to the Hilbert one that have the hereditarily approximation property (shortly hereditarily AP): all their subspaces also have the AP. All these examples were ''sufficiently'' non-symmetric and this fact allows Johnson to ask: whether there exists any Banach space $X$ with symmetric (or, at least, subsymmetric) basis, distinct from the Hilbert space such that each its subspace has the AP? In this paper is shown that there is a Banach space with a subsymmetric basis (non-equivalent to any symmetric one), which enjoys the hered…
We study the Bishop-Phelps-Bollobás property for operators from $\ell_\infty ^4 $ to a Banach space. For this reason we introduce an appropiate geometric property, namely the AHSp-$\ell_\infty ^4$. We prove that spaces $Y$satisfying AHSp-$\ell_\infty ^4$ are precisely those spaces $Y$ such that $(\ell_\infty^4,Y)$ has the Bishop-Phelps-Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, $C_0(L)$ and $L_1 (μ)$ satisfy AHSp-$\ell_\infty ^4 $.
In the article is shown that classes of finite equivalence that are generated by Lebesgue-Riesz spaces Lp have the amalgamation property if and only if p is not equal to an even natural number that is strongly large then 2.
In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a certain type of function constituted a Lyapunov function for some time-invariant linear system. Six years later, it was obtained that this property holds if and only if the Banach space enjoys the self-extension property. However, the knowledge of these spaces needed to be extended in order to make useful this characterization, since there were little information on which classic Banach spaces satisfy this property and its relations with other classic properties. We present the self-extension property in a wider frame, r…
Given a Banach space X and a subspace Y, the pair (X,Y) is said to have the approximation property (AP) provided there is a net of finite rank bounded linear operators on X all of which leave the subspace Y invariant such that the net converges uniformly on compact subsets of X to the identity operator. The main result is an easy to apply dual formulation of this property. Applications are given to three space properties; in particular, if X has the approximation property and its subspace Y is script L-infinity, then X/Y has the approximation property.
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