The dual of the James Tree space is asymptotically uniformly convex

Power type $ξ$-asymptotically uniformly smooth norms

We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $ω$. For every ordinal $ξ$, we characterize the operators, and therefore the Banach spaces, which admit a $ξ$-asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $Sz_ξ(\cdot, \varepsilon)$. We also introduce the $ξ$-Szlenk power type and investigate ideal and factorization properties of classes associated with the $ξ$-Szlenk power type.

Universality, complexity and asymptotically uniformly smooth Banach spaces

For $1<p\le \infty$, we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$-asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by Kalton, Werner and Kurka in the case $p=\infty$.

On asymptotically uniformly smoothness and nonlinear geometry of Banach spaces

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach-Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

加权 Birkhoff 正交 James 型常数

A weighted Birkhoff orthogonal James-type constant

引入加权 Birkhoff 正交 James 型常数 J_λ^⊥(X)，建立其基本性质、与正交 James 常数 J_⊥(X) 的关系，并给出一致非方空间的新刻画。

到局部渐近中点一致凸 Banach 空间中的范数可达算子

Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces

本文证明局部渐近中点一致凸 Banach 空间在含有特定非 ℓ₁ 型基本序列或上 p 估计序列时，必不满足 Lindenstrauss 性质 B。

Existence and higher arity iteration for total asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces

This paper provides a fixed point theorem and iterative construction of a common fixed point for a general class of nonlinear mappings in the setup of uniformly convex hyperbolic spaces. We translate a multi-step iteration, essentially due to Chidume and Ofoedu [4] in such setting for the approximation of common fixed points of a finite family of total asymptotically nonexpansive mappings. As a consequence, we establish strong and del-convergence results which extend and generalize various corresponding results announced in the current literature.

Bourgain--Rosenthal空间的Kadets--Werner修改是渐近中点一致凸的

The Kadets--Werner modification of Bourgain--Rosenthal space is asymptotically midpoint uniformly convex

本文证明Bourgain--Rosenthal空间的Kadets--Werner修改是渐近中点一致凸的，但因其具有Daugavet性质而缺乏连续点性质，从而否定了渐近中点一致凸与渐近一致凸的同构等价性及蕴含连续点性质的两个问题。

The super fixed point property for asymptotically nonexpansive mappings

We show that the super fixed point property for nonexpansive mappings and for asymptotically nonexpansive mappings in the intermediate sense are equivalent. As a consequence, we obtain fixed point theorems for asymptotically nonexpansive mappings in uniformly nonsquare and uniformly noncreasy Banach spaces. The results are generalized for commuting families of asymptotically nonexpansive mappings.

On geometric constants for discrete Morrey spaces

In this note we prove that the $n$-th Von Neumann-Jordan constant and the $n$-th James constant for discrete Morrey spaces $\ell^p_q$ where $1\le p<q<\infty$ are both equal to $n$. This result tells us that the discrete Morrey spaces are not uniformly non-$\ell^1$, and hence they are not uniformly $n$-convex.

一致渐近Hölder-Lipschitz映射的不动点

Fixed points for uniformly asymptotically Hölder-Lipschitz mappings

该研究针对Banach空间中满足特定条件的一致渐近Hölder-Lipschitz映射，证明其在闭凸有界子集上存在不动点，并推广了相关经典定理。

Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces

This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.

Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces

In this note, we extend to the setting of quasi-reflexive spaces a classical result of N. Kalton and L. Randrianarivony on the coarse Lipschitz structure of reflexive and asymptotically uniformly smooth Banach spaces. As an application, we show for instance, that for $1\le q<p$, a $q$-asymptotically uniformly convex Banach space does not coarse Lipschitz embed into a $p$-asymptotically uniformly smooth quasi-reflexive Banach space. This extends a recent result of B.M. Braga.

On strong asymptotic uniform smoothness and convexity

We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov. Our techniques also provide a characterisation of Orlicz functions $M, N$ such that the space of compact operators $\mathcal K(h_M,h_N)$ is asymptotically uniformly smooth. Finally we show that $\mathcal K(X, Y)$ is not strictly …

On geometric properties of Morrey spaces

In this article, we show constructively that Morrey spaces are not uniformly non-$\ell^1_n$ for any $n\ge 2$. This result is sharper than those previously obtained in \cite{GKSS, MG}, which show that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{\rm J}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{\rm NJ}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with $1\le p<q<\infty$ are equal to $n$.

Approximate Carath{é}odory's theorem in uniformly smooth Banach spaces

We study the 'no-dimension' analogue of Carath{é}odory's theorem in Banach spaces. We prove such a result together with its colorful version for uniformly smooth Banach spaces. It follows that uniform smoothness leads to a greedy de-randomization of Maurey's classical lemma \cite{pisier1980remarques}, which is itself a 'no-dimension' analogue of Carath{é}odory's theorem with a probabilistic proof. We find the asymptotically tight upper bound on the deviation of the convex hull from the $k$-convex hull of a bounded set in $L_p$ with $1 < p \leq 2$ and get asymptotically the same bound as in Mau…

Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections

In this paper, a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex and smooth Banach space is proved by using metric projections. This theorem extends and improves the recent strong convergence theorem due to Matsushita and Takahashi [ Appl. Math. Comput. 196 (2008) 422-425] which was established for nonexpansive mappings.

随机半闭原理的一种随机扩展：用于随机渐近非扩张映射

A random demiclosedness principle for random asymptotically nonexpansive mappings

本文基于随机共轭空间与经典共轭空间的内在联系，建立了一个随机半闭原理，将Xu的经典半闭原理从一致凸Banach空间推广到完备的随机一致凸随机范数模。

关于同时零包含性质与内积空间的刻画

On the simultaneous zero inclusion property and characterization of Inner product space

该研究针对光滑且几乎局部一致凸的巴拿赫空间，通过同时零包含性质刻画内积空间，证明满足特定算子条件时Birkhoff-James正交性对称，还解答了相关开放问题。

Best Proximity Point Theorems for Asymptotically Relatively Nonexpansive Mappings

Let $(A, B)$ be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and $T: A\cup B \rightarrow A\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x \in A\cup B$ such that $\|x - Tx\| = \emph{dist}(A, B)$ whenever $T(A) \subseteq B$, $T(B) \subseteq A$. Also, we establish that if $T(A) \subseteq A$ and $T(B) \subseteq B$, then there exist $x \in A$ and $y\in B$ such that $Tx = x$, $Ty = y$ and $\|x - y\| = \emph{dist}(A, B)$. We prove the aforesaid results when the pair $(A, B)$ has the rectangle pr…

Super Asymptotically Nonexpansive Actions of Semitopological Semigroups on Frechet and Locally Convex Spaces

Let LUC$(S)$ be the space of left uniformly continuous functions on a semitopological semigroup $S$. Suppose that $S$ is right reversible and $\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$ be a Fréchet space. Let $τ$ be a locally convex topology of $X$ weaker than the $d$-topology such that the metric $d$ is $τ$-lower semicontinuous. Let $K$ be a $d$--separable and $τ$--compact convex subset of $X$. We show that every jointly $τ$-continuous and super asymptotically $d$-nonexpansive action $S\times K\mapsto K$ of $S$ has a common fixed point. Similar results in the locally conve…

Banach 空间中与等腰正交性相关的对称形式几何常数

Symmetric form geometric constant related to isosceles orthogonality in Banach spaces

本文提出新几何常数 $L_X(t)$ 以等价定义 von Neumann-Jordan 常数，研究其基本性质及与 James 常数等的关系，并据此刻画 Hilbert 空间及 Banach 空间的几何性质。

On Uniformly finitely extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, \emph{On automorphic Banach spaces}, Israel J. Math. 169 (2009) 29--45 and Castillo-Plichko, \emph{Banach spaces in various positions.} J. Funct. Anal. 259 (2010) 2098-2138. We show that they have the Uniform Approximation Property of Pełczyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal --do there exist automorphic spaces other than $c_0(I)$ and $\ell_2(I)$?-- showing that a s…

一致凸重赋范下局部到全局现象的渐近类似物

An asymptotic analog of a local-to-global phenomenon for uniformly convex renormings

本研究针对具有Rolewicz性质$(β)$的Banach空间，给出其存在渐近一致光滑等价范数的无坐标证明，并证明相关模在$(0,1)$内某点为正则存在具性质$(β)$的等价范数，对应局部几何结论的渐近版本。

渐近Hölder非扩张型映射的不动点结果

Fixed point results for mappings of asymptotically Hölder nonexpansive type

将Goebel-Kirk不动点定理推广到渐近Hölder非扩张型映射，通过例子说明新框架严格包含经典情形，并证明若Banach空间包含c₀或ℓ₁的等距同构副本则此类映射的不动点性质失效，最后构造无限维Banach空间中存在无不动点的紧凸集K及其上的渐近Hölder非扩张型仿射自映射T。

On the geometry of the countably branching diamond graphs

In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^ω)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^ω$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^ω)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an …

On Asymptotically Symmetric Banach Spaces

We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c_0 then X is w.a.s.. We obtain an analogous result if c_0 is replaced by …

Banach空间中有限族全渐近非扩张映射强收敛定理的统一

On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces

通过引入新的显式迭代格式统一有限族全渐近I-非扩张映射公共不动点的逼近方法，构造了非渐近非扩张的全渐近非扩张映射实例，并在一致凸Banach空间中证明了强收敛定理。