改进的Tsirelson空间TM上的Wigner定理

设X和Y是赋范空间，如果映射f：X→Y满足{||f（x）+f（y）||，||f（x）-f（y）||}={||x+y||，||x-y||}（x，y∈X），则称f是一个相位等距算子.设g，f：X→Y是映射，若存在相位函数ε：X→{-1，1}，使得ε·f=g，则称g和f是相位等价的.本文将证明改进的Tsirelson空间T_M上的任意满相位等距算子均相位等价于一个线性等距算子.该结论同时也给出了改进的Tsirelson空间T_M上的Wigner型定理.     

Stability Characterizations of ε-isometries on Certain Banach Spaces

Suppose that X, Y are two real Banach Spaces. We know that for a standard ε-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of *. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ε-isometry to be stable in assuming that N is w*-closed in Y*.Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces.We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span[f(X)] contains a complemented linear isometric copy of X; Moreover, if X =Y, then for every e-isometry f: X → X, there exists a surjective linear isometry S:X → X such that f-S is uniformly bounded by 2ε on X.     

单位球面上的等距算子的延拓

本文考虑一般的Banach空间上的等距延拓问题,利用赋范集的概念给出了一些充分条件,使得单位球面间的满等距算子可以延拓为全空间上的线性等距算子。     

On square roots of isometries

Abstract

In various normed spaces we answer the question of when a given isometry is a square of some isometry. In particular, we consider (real and complex) matrix spaces equipped with unitarily invariant norms and unitary congruence invariant norms, as well as some infinite dimensional spaces illustrating the difference between finite and infinite dimensions.     

A Mazur–Ulam theorem in non-Archimedean normed spaces

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.     

Malliavin calculus and decoupling inequalities in Banach spaces

We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided Lp-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.     

The isometrical extensions of 1-Lipschitz mappings on Gteaux differentiability spaces

Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.     

Isometry groups of non standard metric products

We consider isometry groups of a fairly general class of non standard products of metric spaces. We present sufficient conditions under which the isometry group of a non standard product of metric spaces splits as a permutation group into direct or wreath product of isometry groups of some metric spaces.     

The space of left-invariant metrics on a Lie group up to isometry and scaling

We study the spaces of left-invariant Riemannian metrics on a Lie group up to isometry, and up to isometry and scaling. In this paper, we see that such spaces can be identified with the orbit spaces of certain isometric actions on noncompact symmetric spaces. We also study some Lie groups whose spaces of left-invariant metrics up to isometry and scaling are small.     

Ford fundamental domains in symmetric spaces of rank one

We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and domains are essentially subsumed by our work.     

Isometries and additive mapping on the unit spheres of normed spaces

In this paper, we investigate isometric extension problem in general normed space. We prove that an isometry between spheres can be extended to a linear isometry between the spaces if and only if the natural positive homogeneous extension is additive on spheres. Moreover, this conclusion still holds provided that the additivity holds on a restricted domain of spheres.     

Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of Urysohn space

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces U, namely Theorems A and G: Theorem A (Approximation): The group of isometry ISO(U) contains everywhere dense locally finite subgroup; Theorem G (Globalization): For each finite metric space F there exists another finite metric space and isometric imbedding j of F to such that isometry j induces the imbedding of the group monomorphism of the group of isometries of the space F to the group of isometries of space and each partial isometry of F can be extended up to global isometry in . The fact that Theorem G, is true was announced in 2005 by author without proof, and was proved by S. Solecki in [S. Solecki, Extending partial isometries, Israel J. Math. 150 (2005) 315-332] (see also [V. Pestov, The isometry group of the Urysohn space as a Lévy group, Topology Appl. 154 (10) (2007) 2173-2184; V. Pestov, A theorem of Hrushevski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space, math/0702207]) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from [P. Cameron, A. Vershik, Some isometry groups of Urysohn spaces, Ann. Pure Appl. Logic 143 (1-3) (2006) 70-78].     

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.     

Ein beweis des Grauertschen bildgarbensatzes nach ideen von B. Malgrange

In 1960, H. Grauert proved the following coherence theorem [2]: Let X, Y be complex spaces and : X Y a proper holomorphic map. Then, for every coherent analytic sheaf J on X, all direct image sheaves Rⁿ*J are coherent. We give a new proof of this theorem, based on ideas of B. Malgrange. This proof does not use induction on the dimension of the base space Y and can be generalized to relative-analytic spaces X Y where Y belongs to a bigger category of ringed spaces, which contains in particular all complex spaces and differentiable manifolds.     

赋β-范空间中单位球面间的等距算子的线性延拓

本文得到了等距映射的线性延拓的一般结果:设E,F是赋范(或β-严格凸赋β-范)线性空间,若V_0:S_1(E)→S_1(F)是等距,且对任意的x,y∈S_1(E),有‖V_0x-|(?)|V_0y‖≤‖x-|(?)|y‖,(?)∈R,则V_0必可延拓到全空间上等距算子(或线性等距算子)。特别,当E,F是赋范线性空间,V_0是满射或F为严格凸空间时,则V_0必可延拓为全空间的线性等距算子,从而推广了文[3～5]中的相应结果。     

Plasticity in metric spaces

In this paper we examine the properties of EC-plastic metric spaces, spaces which have the property that any noncontractive bijection from the space onto itself must be an isometry.     

How to recognize nonexpansive mappings and isometric mappings

In two real Banach spaces, we shall present two conditions, under one of which each nonexpansive mapping must be an isometry.