Ergodicity and super weak compactness

Applications of the quantification of super weak compactness

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…

演化方程是 $G$-紧的

Evolutionary Equations are $G$-compact

本文在 Picard 意义下证明自治演化方程的 $G$-紧性，通过引入更一般的材料律，去除了空间域有界性、正则性、周期性、遍历性及预解式紧性等假设，并给出仅用静态方程判定时间依赖方程 $G$-收敛的准则。

Nonlinear aspects of super weakly compact sets

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…

Maximal non-compactness of Sobolev embeddings

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…

超自反空间上Lipschitz自由空间中的弱紧性

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。

Representation of measures of noncompactness and its applications related to an initial-value problem in Banach spaces

The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, defined on Banach spaces and its applications. With the aid of a three-time order preserving embedding theorem, we show that for every Banach space $X$, there exist a Banach function space $C(K)$ for some compact Hausdorff space $K$, and an order-preserving affine mapping $\mathbb T$ from the super space $\mathscr B$ of all nonempty bounded subsets of $X$ endowed with the Hausdorff…