Cleon S. Barroso, Donal O'Regan
In this paper, we study a class of Banach spaces, called ϕ-spaces. In a natural way, we associate a measure of weak compactness in such spaces and prove an analogue of Sadovskii fixed point theorem for weakly sequentially continuous maps. A counter-example is given to justify our requirement. As an application, we establish an existence result for a Hammerstein integral equation in a Banach space.
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The purpose of this paper is to provide a proof of James' weak compactness theorem that is able to be taught in a first year graduate class in functional analysis.
To appear in J. Funct. Spaces and Appl.
We introduce an ordinal index which characterizes weak compactness of operators between Banach spaces. We study when classes consisting of operators having bounded index form a closed ideal, the distinctness of the classes, and the descriptive set theoretic properties of this index.
We give new proofs that some Banach spaces have Pełczyński's property $(V)$.
Every absolutely summing linear operator is weakly compact. However, for strongly summing multilinear operators and polynomials - one of the most natural extensions of the linear case to the non linear framework - weak compactness does not hold in general. We show that a subclass of the class of strongly summing multilinear operators/polynomials, sharing its main properties such as Grothendieck's Theorem, Pietsch Domination Theorem and Dvoretzky-Rogers Theorem, has even better properties like weak compactness and a natural factorization 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.
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 prove that a closed convex subset of a Banach space is (super-)weakly compact if and only if it is (super)-ergodic. As a consequence we deduce that super weakly compact sets are characterized by the fixed point property for continuous affine mappings. We also prove that the M-(fixed point property for affine isometries) implies the Banach-Saks property.
We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers, Calderon-Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transform.
We introduce a measure of super weak noncompactness $Γ$ defined for bounded linear operators and subsets in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert generated space. The use of super weak compactness and $Γ$ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos,\linebreak Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits c…
Weak compactness in nice Musielak-Orlicz spaces
本文在满足Δ₂条件的N-函数下，证明了Musielak-Orlicz空间中的弱紧性准则，并展示了该空间具有子序列分裂性质时的弱Banach-Saks性质。
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。
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.
We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1<p,q<\infty$ be such that $1/p+1/q\geq 1$. Let $X$ (resp., $Y$) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower $p$-estimate (resp., $q$-estimate). If $X$ and $Y$ are strongly weakly compactly generated, then so is its projective tensor product $X \widehat{\otimes}_πY$.
We prove that the notions of compactness and weak compactness for a Hankel operator on BMOA are identical.
We obtain exhaustive results and treat in a unified way the question of boundedness, compactness, and weak compactness of composition operators from the Bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like $BMOA$, $Q_α$, and analytic Besov spaces $B^p$. In particular, by combining techniques from both complex and functional analysis, we prove that in this setting weak compactness is equivalent to compactness. For the operators into the corresponding "small" spaces we also characterize the boundedness and show that it is equivale…
Grothendieck's compactness principle for the absolute weak topology
本文在Banach格中建立绝对弱拓扑的格罗滕迪克紧性原理，证明绝对弱紧集与绝对弱序列紧集的等价条件，并刻画绝对弱紧子集可由绝对弱零序列的闭凸包包含当且仅当格具有正Schur性质。
The paper is devoted to the relationship between almost limited operators and weakly compacts operators. We show that if $F$ is a $σ$-Dedekind complete Banach lattice then, every almost limited operator $T:E\rightarrow F $ is weakly compact if and only if $E$ is reflexive or the norm of $F$ is order continuous. Also, we show that if $E$ is a $σ$-Dedekind complete Banach lattice then the square of every positive almost limited operator $ T:E\rightarrow E$ is weakly compact if and only if the norm of $E$ is order continuous.
In this work we prove that if $X$ is a complete locally convex space and $f:X\to \mathbb{R}\cup \{+\infty \}$ is a function such that $f-x^\ast$ attains its minimum for every $x^\ast \in U$, where $U$ is an open set with respect to the Mackey topology in $X^\ast$, then for every $γ\in \mathbb{R}$ and $x^\ast \in U$ the set $\{ x\in X : f(x)- \langle x^\ast , x \rangle \leq γ\}$ is relatively weakly compact. This result corresponds to an extension of Theorem 2.4 in [J. Saint Raymond, Mediterr. J. Math. 10 (2013), no. 2, 927--940]. Directional James compactness theorems are also derived.
Given a parameter dependent fixed point equation $x = F(x,u)$, we derive an abstract compactness principle for the fixed point map $u \mapsto x^*(u)$ under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map $u \mapsto F(x,u)$ is compact for fixed $x$. This result is applied to infinite-dimensional, semi-linear control systems and their reachable sets. More precisely, we extend a non-controllability result of Ball, Marsden, and Slemrod [1] to semi-linear systems. First we consider $L^p$-controls, $p>1$. Subsequently we analyze the case …
Our main result is the following: {\it Let $E$ be a Banach space and $D$ be a weakly compact subset of $E$ with $0\notin D$. If $A$ is a bounded subset of $E$ such that every $x^*\in E^*$ with $x^*(D) >0$ attains its supremum on $A$, then $A$ is weakly relatively compact.}
Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pełczyński's property $(u)$, then either $C$ is weakly compact, contains an $\ell_1$-sequence or a $\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We…
In this paper we show that if $(y_n)$ is a seminormalized sequence in a Banach space which does not have any weakly convergent subsequence, then it contains a wide-$(s)$ subsequence $(x_n)$ which admits an equivalent convex basic sequence. This fact is used to characterize weak-compactness of bounded, closed convex sets in terms of the generic fixed point property ($\mathcal{G}$-$FPP$) for the class of affine bi-Lipschitz maps. This result generalizes a theorem by Benavides, Japón Pineda and Prus previously proved for the class of continuous maps. We also introduce a relaxation of this notion …
In this article, we introduce the notions of sequentially compactness and boundedly compactness in the framework of a newly defined $b_v(s)$-metric space which is a generalization of usual metric spaces and several other abstract spaces. We establish correlations between sequentially compactness and boundedly compactness. Moreover, we prove some fixed point results of contractive mapping in this setting, from which we can deduce several analogous fixed point results. Finally, we illustrate some non-trivial examples to validate the significances and motivations of this manuscript.
Quantitative Schur property and measures of weak non-compactness
本文研究了Banach空间的定量化Schur性质，探讨其与弱非紧性测度的关系，并给出了相关结果和例子。
New Approaches to the Fixed Point Property in L^1 Spaces
本文提出L^1空间中非扩张映射不动点性质的新方法，证明凸集的测度紧致性是充分条件，并统一了超滤积、重赋范等策略，阐明障碍源于子集几何而非空间本身。
We show that in the space of nuclear operators from $\ell^q(Λ)$ to $\ell^p(J)$ the two natural ways of measuring weak non-compactness coincide. We also provide explicit formulas for these measures. As a consequence the same is proved for preduals of atomic von Neumann algebras.
On Weak Compactness and Uniform Regularity in Lipschitz free Spaces
本文研究了Lipschitz自由空间中弱紧性和均匀正则性的性质，通过分析R树的等距映射，提供了弱紧集的内在刻画及范数紧性的新表征。
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…
Weak compactness in Lipschitz-free spaces over superreflexive spaces
本文证明超自反Banach空间上的Lipschitz自由空间中弱预紧子集相对超弱紧，从而得到弱Banach-Saks性质及子空间超自反性等推论，并借助弱序列完备性证明的修改和紧约化方法完成证明。
We introduce two measures of weak non-compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' compactness theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' compactness theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is …
Quantifying (non-)weak compactness of operators on $AL$- and $C(K)$-spaces
研究$AL$-空间间非弱紧算子的表示，证明每个算子在弱紧算子理想中有最佳近似，并通过对偶方法扩展至极不连通$C(L)$-空间。进一步用$\ell_1$上的恒等式因子化刻画$AL$-空间间算子的弱本质范数，得出弱Calkin代数有唯一范数。
Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers
针对非自反Sobolev空间变分问题极小化子存在定理未充分探索的问题，本文证明了$W^{k,1}$中的弱紧性准则（推广Dunford-Pettis定理），并将极小化子存在定理从自反空间推广至非自反空间，范畴论概念助力研究。
The objective of this manuscript is to enquire for the solvability of a specific type of non-linear quadratic integral equations via the interesting notion of measure of non-compactness. Firstly, we inquire into couple of exciting fixed point theorems involving a measure of non-compactness in the setting of a Banach space. Subsequently, bringing into play a suitable measure of non-compactness and the acquired results, we discuss the existence of solution to the aforementioned kind of non-linear quadratic integral equations.
Let $E$ be a Banach space such that $E'$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(μ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^{1}(μ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^{1}(μ,E)$-bounded sequences, which decomposes a subsequence into a relative…
The paper presents some weak compactness criterion for a subset $M$ of the set $\mathfrak{RM}_b(T,\mathcal{G})$ of all positive bounded Radon measures on a Hausdorff topological space $(T,\mathcal{G})$ similar to the Prokhorov criterion for a complete separable metric space. Since for a general topological space the classical space $C_b(T,\mathcal{G})$ of all bounded continuous functions on $T$ can be trivial and so does not separate points and closed sets, instead of $C_b(T,\mathcal{G})$-weak compactness we consider $S(T,\mathcal{G})$-weak compactness with respect to the new uniformly closed …
We continue our study of operator algebras with contractive approximate identities (cais) by presenting a couple of interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series, for example about whether, roughly speaking, `weak compactness' of an operator algebra, or the lack of it, can be seen in the spectra of its elements.
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
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。
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
It has been known that sharp Sobolev embeddings into weak Lebesgue spaces are non-compact but the question of whether the measure of non-compactness of such an embedding equals to its operator norm constituted a well-known open problem. The existing theory suggested an argument that would possibly solve the problem should the target norms be disjointly superadditive, but the question of disjoint superadditivity of spaces $L^{p,\infty}$ has been open, too. In this paper, we solve both these problems. We first show that weak Lebesgue spaces are never disjointly superadditive, so the suggested te…
The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the existence of weak solutions to time fractional PDEs. We first define the weak Caputo derivatives of order $γ\in (0,1)$ for functions valued in general Banach spaces, consistent with the traditional definition if the space is $\mathbb{R}^d$ and functions are absolutely continuous. Based on a Volterra type integral form, we establish some time regularity estimat…
Let $g$ be an analytic function on the unit disc and consider the integration operator of the form $T_g f(z) = \int_0^z fg'\,dζ$. We show that on the spaces $H^1$ and $BMOA$ the operator $T_g$ is weakly compact if and only if it is compact. In the case of $BMOA$ this answers a question of Siskakis and Zhao. More generally, we estimate the essential and weak essential norms of $T_g$ on $H^p$ and $BMOA$.
We consider weighted composition operators on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of weighted composition operators on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of these operators between two different vector-valued Bloch-type spaces. This result appears to be new also in the scalar-valued case.
Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. This paper extends the profile decomposition for Sobolev spaces proved by Solimini (AIHP 1995) to the non-reflexive case p=1. Since existence of concentration profiles relies on weak-star compactness, the corresponding result is set in a larger, conjugate, space of functions of bounded variation. We prove existence of minimizers for related inequalities and generalizations for to spaces of bounded variation on Lie groups.
Weak compactness criteria in $L_1$ space in terms of Orlicz function
本文研究 $L_1$ 空间中弱紧性的 Orlicz 函数刻画，直接证明 K.M. Chong 与 De la Vallée Poussin 准则在 $L_1(0,1)$ 中等价，并讨论其在 $L_1(0,\infty)$ 中的情形。
We prove, among other results, that three standard measures of weak non-compactness coincide in preduals of JBW$^*$-triples. This result is new even for preduals of von Neumann algebras. We further provide a characterization of JBW$^*$-triples with strongly WCG predual and describe the order of seminorms defining the strong$^*$ topology. As a byproduct we improve a characterization of weakly compact subsets of a JBW$^*$-triple predual, providing so a proof for a conjecture, open for almost eighteen years, on weakly compact operators from a JB$^*$-triple into a complex Banach space.
We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let $E$ be a Banach space such that $(B_{E^\ast}, ω^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with \[ \sup(x^\ast,B) < \inf(x^\ast,A) \] attains its infimum on $A$ and its supremum …
Let $ω$ be a continuous weight on $\mathbb R^+$ and let $L^1(ω)$ be the corresponding convolution algebra. By results of Grønbæk and Bade & Dales the continuous derivations from $L^1(ω)$ to its dual space $L^{\infty}(1/ω)$ are exactly the maps of the form $$(D_ϕf)(t)=\int_0^{\infty}f(s)\,\frac{s}{t+s}\,ϕ(t+s)\,ds\qquad\text{($t\in\mathbb R^+$ and $f\in L^1(ω)$)}$$ for some $ϕ\in L^{\infty}(1/ω)$. Also, every $D_ϕ$ has a unique extension to a continuous derivation $\bar{D}_ϕ:M(ω)\to L^{\infty}(1/ω)$ from the corresponding measure algebra. We show that a certain condition on $ϕ$ implies that $\b…
This paper is concerned with the weak solvability of fully nonlinear parabolic variational inequalities with time dependent convex constraints. As possible approaches to such problems, there are for instance the time-discretization method and the fixed point method of Schauder type with appropriate compactness theorems. In this paper, our attention is paid to the latter approach. However, there has not been prepared any appropriate compactness theorem up to date that enables us the direct application of fixed point method to variational inequalities of parabolic type. In order to establish it …
We prove a compactness result for bounded sequences $(u_j)_j$ of functions with bounded variation in metric spaces $(X,d_j)$ where the space $X$ is fixed but the metric may vary with $j$. We also provide an application to Carnot-Carathéodory spaces.
We consider the compactness of derivations from commutative Banach algebras into their dual modules. We show that if there are no compact derivations from a commutative Banach algebra, $A$, into its dual module, then there are no compact derivations from $A$ into any symmetric $A$-bimodule; we also prove analogous results for weakly compact derivations and for bounded derivations of finite rank. We then characterise the compact derivations from the convolution algebra $\ell^1(\Z_+)$ to its dual. Finally, we give an example (due to J. F. Feinstein) of a non-compact, bounded derivation from a un…
We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.
To a generalized tight continuous frame in a Hilbert space $\H$ indexed by a locally compact space $\Si$ endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on $\Si$ in spaces of vectors comparable with $\H$. If the continuous frame is provided by the action of a suitable family of bounded operators on a fixed window, a symbolic calculus emerges, assigning operators in $\H$ to functions on $\Si$. We give some criteria of relative compactness for sets and for families of compact operators, involving tightness properties in terms of objects canonically as…
Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem
本文为具有固定锥形集中Sobolev波前集的分布空间引入局部凸拓扑，推广拉回、紧性、伪微分算子连续性与Rellich引理，并应用于微局域缺陷测度与补偿紧性定理。
We consider operators T : M_0 -> Z and T : M -> Z, where Z is a Banach space and (M_0, M) is a pair of Banach spaces belonging to a general construction in which M is defined by a "big-O" condition and M_0 is given by the corresponding "little-o" condition. Prototype examples of such spaces M are given by $\ell^\infty$, weighted spaces of functions or their derivatives, bounded mean oscillation, Lipschitz-Hölder spaces, and many others. The main result characterizes the weakly compact operators T in terms of a certain norm naturally attached to M, weaker than the M-norm, and shows that weakly …
This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in $\mathrm{L}^p$-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In $\mathrm{L}^p$-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow to quantify the difference in compactness. The definition of MCFs involves a Fouri…
We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia compacta and Radon-Nikodým compacta.
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.
Relative compactness of Itô integrals on the M1 Skorokhod space with identification of limits
该研究在严格一般条件下建立了M1斯科罗霍德空间上伊藤积分序列的弱相对紧性结果，明确了收敛子序列极限点的形式，填补了J1、S拓扑相关的文献空白。
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