Isometric shifts and metric spaces
Jesus Araujo, Juan J. Font
设M是完全度量空间.证明了如果M上的空间或标值有界连续函数允许等距移位，则M是可分的。
Let M be a complete metric space. It is proved that if the space or scalar-valued bounded continuous functions on M admits an isometric shift, then M is separable.
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m-isometric weighted shifts with operator weights
本文研究带算子权重的m-等距加权移位，通过算子系数多项式给出刻画，解决了正权重构造、权重可交换性、完备化问题及伴随算子性质，并提供了相关示例。
Examples and counterexamples of type I isometric shifts
我们给出了C（X）允许I型等距移位的不可分空间$X$的例子，它解决了Gutek { \em et al.} (J.功能Anal. { \bf 101}（1991），97-119）。我们还给出了两个独立的方法获得可分离的例子。第一个允许我们特别是构建的例子无穷多nonhomeomorphic组件在一个子集的希尔伯特空间$\ell^2$。第二个适用于例如与任何n维紧致流形（对于$n \ge 2$）或Sierpienski曲线邻接的序列。这两种技术的结合导致不同的例子，涉及一个收敛序列毗邻的康托尔集：一种方法的情况下，当序列收敛到一个点在康托尔集，和其他一个的情况下，当它收敛外。
A complete solution to the Cauchy dual subnormality problem for torally expansive toral $3$-isometric weighted $2$-shifts
本文通过求解低双次数二变量多项式对应的一对Hausdorff矩问题，完整解决了环面扩张的环面3-等距加权2-移位的柯西对偶次正规性问题。
We characterize metric spaces whose Lipschitz free space is isometric to $\ell_1$. In particular, the Lipschitz free space over an ultrametric space is not isometric to $\ell_1(Γ)$ for any set $Γ$. We give a lower bound for the Banach-Mazur distance in the finite case.
(Almost isometric) local retracts in metric spaces
本文引入度量空间中（几乎等距）局部收缩概念作为 Banach 空间局部补理想的非线性推广，证明了其存在性，并通过有限度量空间间 Lipschitz 函数与几乎等距的可延拓性条件给出刻画。
Higher order isometric shift operator on the de Branges-Rovnyak space
本文证明 de Branges-Rovnyak 空间可作为秩一亏差的扩张拟解析 2n-等距算子的模型空间，并刻画了其上 2n-等距前向移位算子的不变子空间。
On isometric universality of spaces of metrics
本文研究了度量空间的等距普遍性，证明了具有丰富子空间的可度量空间上的有界度量空间和度量空间分别对特定类别的度量空间具有普遍性，并给出了紧致可数空间下的反例。
Note on almost isometric ideals and local retracts in Banach and metric spaces
本文提出了一种新方法，证明了非分离Banach空间中存在大量几乎等距理想和度量空间中大量几乎等距局部再退化，同时展示了在非分离Banach空间的每个对偶空间上存在大量非平凡投影。
Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $\ell_\infty$.
In this paper, we provide an infinite metric space $M$ such that the set $\mbox{SNA}(M)$ of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to $c_0$. This answers a question posed by Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that $\mbox{SNA}(M)$ contains an isometric copy of $c_0$ whenever $M$ is a metric space which is not uniformly discrete. In particular, the latter holds true for infinite compact metric spaces while it does not for proper metric spaces. Some positive results in…
Main results: (a) If a metric space contains $2n$ elements, the transportation cost space on it contains a $1$-complemented isometric copy of $\ell_1^n$. (b) An example of a finite metric space whose transportation cost space contains an isometric copy of $\ell_\infty^4$. Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces.
Isometric copies of $\ell_\infty^n$ in Lipschitz-free spaces over finite metric spaces
针对每个自然数n，构造有限度量空间M_n，使其Lipschitz-free空间包含ℓ_∞^n的等距拷贝，从而解决Khan等人提出的问题，并刻画可嵌入的有限维空间为多面体空间。
The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. The TC spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein spaces. A new notion of a roadmap pertinent to a transportation problem on a finite metric space has been introduced and used to simplify proofs for the results on representation of TC spaces as quotients of $\ell_1$ spaces on the edge set over the cycle space. A Tolstoi-type theorem for roadmaps is proved, and directed subgraphs of the canonical graphs, which are supports of maximal optimal …
A Toeplitz operator $T_φ$, $φ\in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $φ_1, φ_2 \in H^\infty(\mathbb{D}^n)$ such that $φ_1$ and $φ_2$ depends on different variables and $φ= \barφ_1 φ_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry …
We prove that the Lipschitz-free space over a countable proper metric space is isometric to a dual space and has the metric approximation property. We also show that the Lipschitz-free space over a proper ultrametric space is isometric to the dual of a space which is isomorphic to c_0.
The Cauchy dual subnormality problem in two variables
研究解析左逆交换对环面 2 - 等距算子，满足联合核条件时与可逆权重算子值加权移位酉等价，初始权重交换时柯西对偶联合次正规，还构造反例说明初始权重不交换时柯西对偶非联合次正规。
On perturbations of the isometric semigroup of shifts on the semiaxis
本文研究了移位半群$(τ_t)_{t\ge 0}$在$L^2(\R_+)$上的扰动$(\tildeτ_t)_{t\ge 0}$，其性质是$\tildeτ_t - τ_t$属于具有$p\ge 1$的Schatten-von Neumann类$\gS_p$.证明了对于半群$(\tildeτ_t)_{t\ge 0}$的余生成元的Wold-Kolmogorov分解中的酉分量，通过$\gS_1$扰动可以得到{ \it any singular}谱型.基于Hardy空间$H^2$的模型空间理论，我们给出了一个具有给定谱型的扰动的显式构造。我们还证明了，通过对具有$p>1$的类$\gS_p$的扰动，可以得到扰动半群的酉分支的{ \it any}指定谱型.
A l_1-predual which is not isometric to a quotient of C(alpha)
大约20年前，Johnson和Zippin证明了每个可分的L_1（mu）-预对偶都等距于C（Delta）的商，这里Delta是康托尔集。本文将证明l_1-predictor定理的自然类比不成立。我们将证明：对于可数紧度量空间，有许多l_1-预紧空间不等距于任何C（K）-空间的商。证明了当K是可数紧度量空间时，C（K）-空间的l_1-预紧性与预紧性之间关系的一些一般性结果。本文的结果在研讨会上提出的巴拿赫空间理论在梅里达，委内瑞拉，1992年1月。
The aim of this paper is to study the Wold-type decomposition in the class of $m$-isometries. One of our main results establishes an equivalent condition for an analytic $m$-isometry to admit the Wold-type decomposition for $m\ge2$. In particular, we introduce the $k$-kernel condition which we use to characterize analytic $m$-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $m\ge2$. As a result, we also show that $m$-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to…
Rigidity of AMN vector spaces
度量向量空间是渐近度量可赋范的（AMN），如果存在一个范数渐近等距于距离。证明了AMN向量空间在渐近等距扰动下是度量向量空间类中的刚性空间。这个结果来自于一个一般的度量可范性准则。如果距离是平移不变的，并且满足一个近似乘法条件，则存在Lipschitz等价范数。进一步，我们给出了距离渐近等距于范数的充要条件。
We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.
Isometric embeddings of compact spaces into Banach spaces
证明了紧致度量空间$K$的存在性，使得只要$K$等距嵌入到巴拿赫空间$Y$中，则任何可分巴拿赫空间线性等距于$Y$的子空间.我们还解决了以下相关的问题：如果一个巴拿赫空间$Y$包含一个等距副本的单位球或一些特殊的紧子集的可分巴拿赫空间$X$，它是否一定包含一个子空间等距$X$？当$X$是多面体有限维空间$c_0$或$\ell_1$时，我们肯定地回答这个问题。
Sobolev mappings: Lipschitz density is not an isometric invariant of the target
若$M$是紧致光滑流形，$X$是紧致度量空间，则索伯列夫空间$W^{1,p}(M,X)$是通过将$X$等距嵌入到巴拿赫空间中来定义的.证明了Lipschitz映射${\rm Lip}\,(M,X)$在$W^{1,p}(M,X)$中是否稠密的问题的答案可能依赖于目标的等距嵌入.
Global and local boundedness of Polish groups
我们提出了一个全面的理论，波兰群的有界性的发展，主要集中在Roelcke预紧性（下一致性的预紧性）和性质（OB）（可分度量空间上的所有等距作用的有界性）。特别是，这些属性的特点是轨道结构的等距行动的度量空间和等距或连续仿射表示可分离的巴拿赫空间。
Given two metric spaces $M$ and $N$ we study, motivated by a question of N. Weaver, conditions under which an isometric composition operator $C_ϕ:\mathrm{Lip}_0(M)\longrightarrow \mathrm{Lip}_0(N)$ is isometric depending on the properties of $ϕ$. We obtain a complete characterisation of those operators $C_ϕ$ in terms of a property of the function $ϕ$ in the case that $B_{\mathcal F(M)}$ is the closed convex hull of its preserved extreme points. Also, we obtain necessary and sufficient conditions for $C_ϕ$ being isometric in the case that $M$ is geodesic.
On Dirichlet-type and $n$-isometric shifts in finite rank de~Branges--Rovnyak spaces
本文研究有限秩 de Branges–Rovnyak 空间中 Dirichlet 型函数空间上的移位算子，证明有理符号与有限原子测度及有限秩亏格算子的等价性，并刻画了多参数情形下扩张移位算子的允许元组条件。
Given pointed metric spaces $X$ and $Y$, we characterize the basepoint-preserving Lipschitz maps $ϕ$ from $Y$ to $X$ inducing an isometric composition operator $C_ϕ$ between the Lipschitz spaces $Lip_0(X)$ and $Lip_0(Y)$, whenever $X$ enjoys the peak property. This gives an answer to a question posed by N. Weaver in his book [Lipschitz algebras. Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018].
Free Spaces over Countable Compact Metric Spaces
证明了可数紧度量空间上的Lipschitz-free空间与对偶空间等距，并具有度量逼近性质.
Best constants for Lipschitz embeddings of metric spaces into c_0
我们回答了一个问题Aharoni表明，每一个可分度量空间可以Lipschitz 2-嵌入到$c_0$，这一结果是尖锐的，这提高了早期估计Aharoni，Assouad和Pelant。我们使用我们的方法来检查Lipschitz嵌入到$c_0$的经典$\ell_p-$空间的最佳常数，并给出其他应用。证明了若巴拿赫空间几乎等距嵌入$c_0$，则它线性几乎等距嵌入$c_0$ .我们还研究了Lipschitz嵌入到$c_0^+$中。
The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of $\ell_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into…
Banach's Isometric Conjecture over the Complex Field
该研究完成复域巴拿赫等距猜想证明，还证四元数域对应结论，用Lu和Yang的丛次数机制完成核心几何论证，且将结果推广到多类空间与度量。
Isometric rigidity of the Wasserstein space over the plane with the maximum metric
本文研究了配备最大范数度量的分支空间上的 $p$-Wasserstein 空间，证明了对所有 $p\geq1$ 这些空间均具有等距刚性，并揭示了其与欧氏度量及一维情况在刚性上的显著差异。
On quantum symmetries of compact metric spaces
紧量子群在紧度量空间$(X,d)$上的作用是（D）-等距的，如果距离函数被$X\times X$上的对角作用保持.我们证明了在这种意义下的等距作用具有以下附加性质：量子群上的概率测度的卷积半群对$X$上的连续函数代数的相应作用收缩Lipschitz常数.换句话说，由于H，它在另一种意义上是等距的。Li，J. Quaegebeur和M. Sabbe;这部分地回答了D.哥斯瓦米我们还介绍了其他可能的等距量子作用的概念，在Wasserstein $p$ -$X$的概率测度之间的距离$p\ge 1$，广泛用于最佳运输。事实证明，所有这些量子等距的定义都符合一个暗示的塔，上面两个在塔的末端。我们推测它们都是等价的。
We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for $p\in [1,\infty]$, every proper subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and only if $X$ contains uniformly the $\ell_p^n$'s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.
This work aims to establish new results pertaining to the structure of transportation cost spaces. Due to the fact that those spaces were studied and applied in various contexts, they have also become known under different names such as Arens-Eells spaces, Lipschitz-free spaces, and Wasserstein spaces. The main outcome of this paper states that if a metric space $X$ is such that the transportation cost space on $X$ contains an isometric copy of $\ell_1$, then it contains a $1$-complemented isometric copy of $\ell_1$.
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