Pablo Galindo, V. C. C. Miranda
A wide new class of subsets of a Banach space $X$ named coarse $p$-limited sets ($ 1\leq p < \infty$) is introduced by considering weak* $p$-summable sequences in $X'$ instead of weak* null sequences. We study its basic properties and compare it with the class of compact and weakly compact sets. Results concerning the relationship of coarse $p$-limited sets with operators are obtained.
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A boundary for a Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. Godefroy's Boundary Problem asks whether nevertheless both topologies have the same bounded compact sets. This paper contains the answer in the positive.
An example of an infinite dimensional and separable Banach space is given, that is not isomorphic to a subspace of l1 with no infinite equilateral sets.
Non dentable sets in Banach spaces with separable dual
构造了一个非RNP巴拿赫空间E，使得$E^{*}$是可分的，且RNP在E的子集上等价于PCP.
In this paper we study different aspects of the representation of weak*-compact convex sets of the bidual $X^{**}$ of a separable Banach space $X$ via a nested sequence of closed convex bounded sets of $X$.
We introduce and study the notion of overcomplete set in a Banach space, that subsumes and extends the classical concept of overcomplete sequence in a (separable) Banach space. We give existence and non-existence results of overcomplete sets for a wide class of (non-separable) Banach spaces and we study to which extent properties of overcomplete sequences are retained by every overcomplete set.
We introduce the wave-front set for distributions with respect to Fourier images of weighted translation invariant Banach function spaces. We prove that usual mapping properties for pseudo-differential operators hold in the context of such wave-front sets.
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…
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.
We show that every infinite dimensional Banach space has a closed and bounded convex set that is not remotal.
Quantification of Banach-Saks properties of higher orders
本文研究高阶Banach-Saks集与弱Banach-Saks集的量化方法，证明相关刻画与紧性关系的定量版本，引入新的弱紧性度量，并通过例子说明量化的局限性。
the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible, admits an equivalent Frechet differentiable norm.
In the present article we prove a fixed point theorem for reflections of compact convex sets and give a new characterization of state space of JB-algebras among compact convex sets. Namely they are exactly those compact convex sets which are strongly spectral and symmetric.
Reciprocal Dunford--Pettis sets and V$^*$-sets in Banach lattices
本文证明 Banach 格中有界集的 V$^*$-集、互反 Dunford–Pettis 集与不交弱紧性三者等价。
It is shown that every set I(m) of Banach lattices of measurable functions defined on a measure space (Q,S,m), equipped with a some natural ordering became a modular lattice, which is Dedekind complete provided m is a probability measure. Moreover, some natural operations on considered spaces are in Galois connexion.
Let $X$ be an infinite dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exists a constant $λ>0$ and an infinite sequence $(x_i)_{i=1}^\infty\subset X$ such that $\|x_i-x_j\|=λ$ for all $i\neq j$.
In this paper, we unify the theory of SSD spaces and the theory of strongly representable sets, and we apply our results to the theory of the various classes of maximally monotone sets. In particular, we prove that type (ED), dense type, type (D), type (NI) and strongly representable are equivalent concepts and, consequently, that the known properties of strongly representable sets follow from known properties of sets of type (ED).
In this note we introduce the notion of $t$-analytic sets. Using this concept, we construct a class of closed prime ideals in Banach function algebras and discuss some problems related to Alling's conjecture in $H^\infty$. A description of all closed $t$-analytic sets for the disk-algebra is given. Moreover, we show that some of the assertions in Daoui et al. (Proc. Am. Math. Soc. 131:3211-3220, 2003) concerning the $O$-analyticity and $S$-regularity of certain Banach function algebras are not correct. We also determine the largest set on which a Douglas algebra is pointwise regular.
Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a $C^*$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we will prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions.
Haar null sets were introduced by J.P.R. Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, U.B. Darij defined a categorical version of Haar null sets, which he named Haar meagre sets. The present paper aims to show that, whenever $C$ is a closed, convex subset of a separable Banach space, $C$ is Haar null if and only if $C$ is Haar meagre. We then use this fact to improve a theorem of E. Matoušková and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation …
In this note we study the endomorphisms of certain Banach algebras of infinitely differentiable functions on compact plane sets, associated with weight sequences M. These algebras were originally studied by Dales, Davie and McClure. In a previous paper this problem was solved in the case of the unit interval for many weights M. Here we investigate the extent to which the methods used previously apply to general compact plane sets, and introduce some new methods. In particular, we obtain many results for the case of the closed unit disc. This research was supported by EPSRC grant GR/M31132
The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if and only if $X$ is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. {\bf (2)} Let $X$ be a non-reflexive Banach space. Then there exists a convex subset $A\subset X^*$ such that $A^{(1)}\neq {\bar{A}\,}^*$ (the latter denotes the weak$^*$ closure of $A$). {\bf (3)} Let $X$ be a quasi-reflexive Banach space and…
Recently, J.X. Chen et al. introduced and studied the class of almost limited sets in Banach lattices. In this paper we establish some characterizations of almost limited sets in Banach lattices (resp. wDP* property of Banach lattices), that generalize some results obtained by J.X. Chen et al.. Also, we introduce and study the class of the almost limited operators, which maps the closed unit bull of a Banach space to an almost limited subset of a Banach lattice. Some results about the relationship between the class of almost limited operators and that of limited (resp. L- and M-weakly compact,…
In his study of the Radon Nikodým property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set $A$ that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set $A$ (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of $A$-valued quasimartingales with sharply divergent behavior.
W^*-derived sets of transfinite order of subspaces of dual Banach spaces
这是一份英文翻译的文件最初发表在俄罗斯和乌克兰在1987年。在附录中他的书S.巴拿赫介绍了以下定义让$X$是一个巴拿赫空间和$Γ$是一个子空间的对偶空间$X^*$。$Γ$中$w^{*}$ -收敛序列的所有极限的集合称为$Γ$的$w^*$ { \it -导出集}，记为$Γ_{(1)}$ .对于序数$α$，$w^{*}$ -{ \it导出序集} $α$由等式归纳定义：$$ Γ_{(α)}=\bigcup _{β<α}((Γ_{(β)})_{(1)}. $$
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 $Y$.
Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space $E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a continuous function $f:\X\to A$ such that (1) $A$ is generated by $f(\X)$, (2) the character space of $A$ is homeomorphic to $K$, and (3) $K=\vsp(f)$ the joint spectrum of $f$. In case $E=\c(X)$, where $X$ is a compact Hausdorff space, we will see that $\X$ can be replaced by $X$.
Classes of Banach spaces that are finitely, strongly finitely or elementary equivalent are introduced. On sets of these classes topologies are defined in such a way that sets of defined classes become compact totally disconnected topological spaces. Results are used in the problem of synthesis of Banach spaces, and to describe omittable spaces that are defined below.
On vector-valued inequalities for Sidon sets and sets of interpolation
设$E$是整数的Sidon子集，$X$是巴拿赫空间.然后皮西尔已表明，$E$ -谱多项式的值在$X$行为就像Rademacher和关于$L_p-$规范。我们考虑了当$X$是拟巴拿赫空间时的情形。对于一般的拟巴拿赫空间我们证明了一个类似的结果成立当且仅当$E$是一个插值集（$I_0$ -集）.然而，对于某些特殊类的准巴拿赫空间，我们能够证明这样的结果，为更大的集。因此，如果$X$被限制为“自然”，则结果对所有西顿集都成立。我们还考虑空间与pluisubharmonic规范，并介绍了类解析西顿集。
We introduce global wave-front sets $\operatorname{WF}_{\mathcal B} (f)$, $f\in {\mathscr S}^\prime(\textbf{R}^d)$, with respect to suitable Banach or Fréchet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(ω,\mathscr B)$, where $ω$ is an appropriate weight function and $\mathscr B$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to $\operatorname{WF}_{\mathcal B} (f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally …
In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w^*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications.
We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin sigma-P-porous sets where "P" can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Suslin cone small set in Asplund spaces. As an application we prove a theorem stating that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set.
In this paper, we define a Banach SNL space to be a Banach space with a certain kind of linear map from it into its dual, and we develop the theory of linear $L$-positive subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis-Browder theorem.
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.
d-Operators in Banach Lattices
本文研究Banach格中把不交有界集映为紧、弱紧及有限集的d-算子，发现其与经典算子行为不同，并引入d-Andrews算子，将其映射到Dunford-Pettis集。
Disjointly Weak Compactness in Banach Lattices
本文刻画 Banach 格中的不交弱紧集，并据此证明从 Banach 空间到 Banach 格的有界线性算子为 almost (L)-limited 算子当且仅当它是不交弱紧算子，同时给出其弱预紧性与弱紧性结果。
In this paper we study properties of complete sets and of completions of sets in Banach spaces. We consider the family of completions of a given set and its size; we also study in detail the relationships concerning diameters, radii, and centers. The results are illustrated by several examples.
We investigate weak$^*$ derived sets, that is the sets of weak$^*$ limits of bounded nets, of convex subsets of duals of non-reflexive Banach spaces and their possible iterations. We prove that a dual space of any non-reflexive Banach space contains convex subsets of any finite order and a convex subset of order $ω+ 1$.
A note on the Lumer--Phillips theorem for bi-continuous semigroups
本文在 Banach 空间附加更粗 Hausdorff 局部凸拓扑的情形下，利用半范数有向系的耗散性与值域条件，给出了 $τ$-双连续半群生成元的 Lumer–Phillips 型刻画。
The paper is devoted to the convex-set counterpart of the theory of weak$^*$ derived sets initiated by Banach and Mazurkiewicz for subspaces. The main result is the following: For every nonreflexive Banach space $X$ and every countable successor ordinal $α$, there exists a convex subset $A$ in $X^*$ such that $α$ is the least ordinal for which the weak$^*$ derived set of order $α$ coincides with the weak$^*$ closure of $A$. This result extends the previously known results on weak$^*$ derived sets by Ostrovskii (2011) and Silber (2021).
We prove the optimal estimate between the Szlenk and $w^*$-dentability indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^*$-dentability index of a Banach space and of an operator.
Let $A$ be a Banach algebra. For $f\in A^{\ast}$, we inspect the weak sequential properties of the well-known map $T_f:A\to A^{\ast}$, $T_f(a) = fa$, where $fa\in A^{\ast}$ is defined by $fa(x) = f(ax)$ for all $x\in A$. We provide equivalent conditions for when $T_f$ is completely continuous for every $f\in A^{\ast}$, and for when $T_f$ maps weakly precompact sets onto L-sets for every $f\in A^{\ast}$. Our results have applications to the algebra of compact operators $K(X)$ on a Banach space $X$.
We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets.
This paper is devoted to the study of the relatively compact sets in Quasi-Banach function spaces, providing an important improvement of the known results. As an application, we take the final step in establishing a relative compactness criteria for function spaces with any weight without any assumption.
We define the ($w^\ast$-) boundedness property and the ($w^\ast$-) surjectivity property for sets in normed spaces. We show that these properties are pairwise equivalent in complete normed spaces by characterizing them in terms of a category-like property called ($w^\ast$-) thickness. We give examples of interesting sets having or not having these properties. In particular, we prove that the tensor product of two $w^\ast$-thick sets in $\Xastast$ and $\Yast$ is a $w^\ast$-thick subset in $L(X,Y)^\ast$ and obtain as a concequense that the set $w^\ast -exp\:B_{K(l_2)^\ast}$ is $w^\ast$-thick.
Let $X$ be a Banach space and $Γ\subseteq X^*$ a total linear subspace. We study the concept of $Γ$-integrability for $X$-valued functions $f$ defined on a complete probability space, i.e. an analogue of Pettis integrability by dealing only with the compositions $\langle x^*,f \rangle$ for $x^*\in Γ$. We show that $Γ$-integrability and Pettis integrability are equivalent whenever $X$ has Plichko's property ($\mathcal{D}'$) (meaning that every $w^*$-sequentially closed subspace of $X^*$ is $w^*$-closed). This property is enjoyed by many Banach spaces including all spaces with $w^*$-angelic dual…
We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.
A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist non-separable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of $\ell_1([0,1])$. A wider class of renormings of $\ell_1([0,1])$ which admit no uncountable equilateral sets is also considered.
The paper concerns the problem whether a nonseparable $\C(K)$ space must contain a set of unit vectors whose cardinality equals to the density of $\C(K)$ such that the distances between every two distinct vectors are always greater than one. We prove that this is the case if the density is at most continuum and we prove that for several classes of $\C(K)$ spaces (of arbitrary density) it is even possible to find such a set which is $2$-equilateral; that is, the distance between every two distinct vectors is exactly 2.
There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \in…
Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the $c_0$-saturation of $C(K)$, $K$ countable compacta. The second one is the construction of weakly-null sequences generalizing the example of Maurey-Rosenthal.
We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelof number in the weak topology and other related questions.
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.
On compact (limited) operators between Hilbert and Banach spaces
本文研究 Hilbert 到 Banach 空间间线性算子集合的紧性，给出集体紧（有限）集的正交刻画。
$DW$-DP operators and $DW$-limited operators on Banach lattices
本文定义了Banach格上与不交弱紧集相关的$DW$-DP和$DW$-限制算子，刻画其等价算子类特征，并给出正算子的逼近性质，拓展了Banach格算子理论研究。
This paper deals with the \emph{Bishop-Phelps-Bollobás property} (\emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$, not just on its closed unit ball $B_X$. We firstly prove that the \emph{BPBp} holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces $X$ and $Y$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed convex subset $D$ of $X$, and also that for a Banach space $Y$ with property $(β)$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed ab…
We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincaré inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.
$DW$-compact operators on Banach lattices
本文研究 Banach 格上的 $DW$-紧算子，证明其等价于既是 Dunford-Pettis 又是 $AM$-紧的算子，并以此刻画了特定 Banach 格的性质。
We study existence and partial regularity relative to the weighted Steiner problem in Banach spaces. We show $C^1$ regularity almost everywhere for almost minimizing sets in uniformly rotund Banach spaces whose modulus of uniform convexity verifies a Dini growth condition.
We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal equilateral set in C(K) is countable.
Free objects in Banach space theory
本文综述了与Banach空间相关的三类自由对象（Lipschitz自由空间、全纯自由空间和自由Banach格）的最新进展，并重点探讨了这些发展之间的相似性。
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