COMPOSITION OPERATORS ON ANALYTIC VECTOR-VALUED NEVANLINNA CLASSES

Let φ be an analytic self-map of the complex unit disk and X a Banach space. This paper studies the action of composition operator Cφ： f→foφ on the vector-valued Nevanlinna classes N（X） and Na（X）. Certain criteria for such operators to be weakly compact are given. As a consequence, this paper shows that the composition operator Cφ is weakly compact on N（X） and Na（X） if and only if it is weakly compact on the vector-valued Hardy space H^1 （X） and Bergman space B1（X） respectively.     

K-POTENT PRESERVING LINEAR MAPS

Let B(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let k ≥ 2 be an integer and φ a weakly continuous linear surjective map from B(X) into itself. It is shown that φ is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number λ satisfying λk-1= 1. Let A be a von Neumann algebra and B be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multiplied by an invertible element with (k - l)-th power I.     

The Asymptotic Average Shadowing Property and Its Applications

Let （X, d） be a bounded metric space and f ： X → X be a uniformly continuous surjection. For a given dynamical system （X, f） which may not be compact, we investigate the relation between the asymptotic average shadowing property（AASP）, transitivity and mixing. If f has the AASP, then the following statements hold： （1） f n is chain transitive for every positive integer n; （2） If X is compact and f is an expansive homeomorphism, then f is topologically weakly mixing; （3） If f is equicontinuous, then f is topologically weakly mixing; （4） If X is compact and f is equicontinuous, then f ×f is a minimal homeomorphism. We also show that the one-sided shift map has the AASP and the identity map 1 X does not have the AASP. Furthermore, as its applications, some examples are given.     

On Universally Left-stability of ε-Isometry

LetX,Y be two real Banach spaces andε≥0.A map f:X→Y is said to be a standardε-isometry if|f(x)f(y)x y|≤εfor all x,y∈X and with f(0)=0.We say that a pair of Banach spaces(X,Y)is stable if there existsγ0 such that,for every suchεand every standardε-isometry f:X→Y,there is a bounded linear operator T:L(f)≡spanf(X)→X so that T f(x)x≤γεfor all x∈X.X(Y)is said to be universally left-stable if(X,Y)is always stable for every Y(X).In this paper,we show that if a dual Banach space X is universally left-stable,then it is isometric to a complemented w-closed subspace of∞(Γ)for some setΓ,hence,an injective space;and that a Banach space is universally left-stable if and only if it is a cardinality injective space;and universally left-stability spaces are invariant.     

The central limit theorem and chaos

Let X be a compact metric space studies some relationships between stochastic and f ： X→ X be a continuous map. This paper and topological properties of dynamical systems. It is shown that if f satisfies the central limit theorem, then f is topologically ergodic and / is sensitively dependent on initial conditions if and only if / is neither minimal nor equicontinuous.     

关于Banach空间中渐近非扩张型半群的不动点与渐近行为

Let X be a Banach space with a weakly continuous duality map Jφ,C a non-empty weakly compact convex subset of X, and T:(T(t):t∈S} an asymptotically nonexpansive type semigroup on C. In this paper, the inequality K∩F(T)≠0 is characterized, where K is a subset of C and F(T) is the set of all common fixed points of T. Furthermore, it is shown that an almost-orbit {u(t):t∈S} of T converges weakly to a point in F(T) if and only if {u(t):t∈S}is weakly asymptotically regular.     

Linear Maps Preserving Idempotence on Nest Algebras

In this paper, we discuss the rank-l-preserving linear maps on nest algebras of Hilbertspace operators. We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear hijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism.     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Topological Anosov Maps of Non-compact Metric Spaces

Let X be a metric space. We say that a continuous surjection f: X→X is a topological Anosov map (abbrev. TA-map) if f is expansive and has pseudo-orbit tracing property with respect to some compatible metric for X. This paper studies the properties of TA-maps of non-compact metric spaces and gives some conditions for the map to be topologically mixing.     

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

逐点伪轨跟踪性质及其应用

本文给出紧致度量空间逐点伪轨跟踪性质的定义,该定义是伪轨跟踪性质定义的推广.作为应用,证明如下结论:(i)若f具有逐点伪轨跟踪性质,且对任意k∈Z ,fk为链转换的,那么对任意k∈Z ,fk为开集转换;(ii)若f具有逐点伪轨跟踪性质,且对任意n∈Z ,fn为链转换的,则f具有初始敏感依赖性质;(iii)若f为开集混合的,且具有逐点伪轨跟踪性质,那么f具有性质P;(iv)设f:(X,d)→(X,d)是同胚映射,那么f具有逐点伪轨跟踪性质当且仅当移位映射σf具有逐点伪轨跟踪性质.     

Devaney混沌的随机性质

讨论了Devaney混沌的随机性质,证明了如果度量空间上的连续变换f是弱混合的,那么f是拓扑传递的,并且若f的周期点稠密,则f还是初值敏感的.     

Topological weak mixing and quasi-Bohr systems

A minimal dynamical system (X, T) is called quasi-Bohr if it is a non-trivial equicontinuous extension of a proximal system. We show that if (X, T) is a minimal dynamical system which is not weakly mixing then some minimal proximal extension of (X, T) admits a nontrivial quasi-Bohr factor. (In terms of Ellis groups the corresponding statement is:AG′=G implies weak mixing.) The converse does not hold. In fact there are nontrivial quasi-Bohr systems which are weakly mixing of all orders. Our main tool in the proof is a theorem, of independent interest, which enhances the general structure theorem for minimal systems.     

A Characterization of Topologically Transitive Attributes for a Class of Dynamical Systems

In this work,by virtue of the properties of weakly almost periodic points of a dynamical system (X,T) with at least two points,the authors prove that,if the measure center M(T) of T is the whole space,...     

On strictly weakly mixing <Emphasis Type="Italic">C</Emphasis>*-dynamical systems

We consider strictly ergodic and strictly weakly mixing C*-dynamical cystems. We establish that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. We also investigate some weighted uniform ergodic theorem with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems.     

Ergodic and Mixing Probability Measures on [SIN] Groups

We prove that the concepts of completely mixing, mixing, and weakly mixing probability measures on a locally compact group in the class [SIN] are mutually equivalent. A probability measure on such a group is completely mixing if and only if it is ergodic and aperiodic. For measures that are not necessarily aperiodic ergodicity is shown to be equivalent to a mixing-like property which we call almost mixing. Some results on when a probability measure on a [SIN] group is ergodic are also developed. Our methods are based on the theory of boundaries of random walks.     

大偏差定理与初值敏感性

本文讨论了动力系统的统计性质和动力性质的某些关系.对于紧致度量空间X上的连续自映射f,我们证明了:如果f满足大偏差定理,那么f是初值敏感的当且仅当f不是极小等度连续的.     

按序列分布混沌与拓扑混合

本文讨论了按序列分布混沌与拓扑混合的关系,并证明了:若X为至少两点的可分局部紧致度量空间,连续映射f:X→X是拓扑混合的,则对于任一正整数递增序列{mi},存在X的c-稠密Fσ子集D是f按{mi}的某子序列的分布混沌集.