HYPERCYCLICITY, SUPERCYCLICITY AND GENERALIZED RAKOEVIC'S PROPERTY

A Banach space operator satisfies generalized RakoSevi5＇s property （gw） if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property （gw）.

Crystal算子的Hypercyclicity

称复可分Banach空间上的算子T是Crystal算子,如果T在其任意非零不变子空间上的限制相似于T。对于某个x∈X,称T是超循环的(Hypercyclic),若{x,Tx,T2x,…,Tnx,…}=X;若∨{Tnx:n≥0}=X,则称T是循环的;称T是严格循环的,如果存在x∈X使得A(T)x=X,其中A(T)是T生成的闭子代数。研究Crystal算子T的循环性:1.Crystal算子都是循环算子,但不是严格循环的;2.利用向量值解析函数的性质,证明:当1∈σ(T)\σe(T)时,T是超循环算子。

On the Supercyclicity and Hypercyclicity of the Operator Algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.

加权空间l^2(Z,a)上左移位算子的超循环性

给出双边加权左移位算子BW在加权空间l2(Z,a)上是hypercyclic的一个充要条件(定理1),应用定理1的讨论方法,得到BW和λB是l2(Z,a)上的supercyclic算子的充要条件.再考虑加权空间上的算子B和BW的hypercyclic向量,给出了加权空间上移位算子的hypercyclic向量与权序列a={an}n∈Z之间存在的一些关系.

RECURRENCE FOR WEIGHTED TRANSLATIONS ON GROUPS

Let G be a locally compact group, and let 1≤p〈∞. We characterize topolog- ically multiply recurrent weighted translation operators on LP（G） in terms of the Haax mea- sure and the weight function. We also show that there do not exist any recurrent weighted translation operators on L^∞ （G）.

Hypercyclic与Supercyclic的Toeplitz算子

首先运用函数论的方法,阐述了在Hardy空间以及Bergman空间上,当符号φ满足某种条件时,余解析Toeplitz算子Tφ为Hypercyclic或Supercyclic算子.其次运用谱的知识及指标理论,阐述了当符号φ满足某种条件时,Toeplitz算子Tφ位于Hc(H)或Sc(H)中.

算子的Supercyclic性

依据判定一列算子是Hypercyclic的充要条件,得到一种判定算子是Supercyclic的标准,并将该标准应用到权序列均非零双侧加权移位算子上,得到了权序列均非零双侧加权移位算子满足该标准的充要条件.

WEYL'S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions： （i） If T or T＊ ∈ SC, then generalized a- Browder＇s theorem holds for f（T） for every f ∈ Hol（σ（T））. （ii） If T or T＊ ∈ HC has topological uniform descent at all λ ∈ iso（a（T））, then generalized Weyl＇s theorem holds for f（T） for every f ∈ Hol（σ（T））. （iii） If T ∈ HC has topological uniform descent at all λ ∈（T）, then T satisfies generalized Weyl＇s theorem. （iv） Let T ∈ HC. If T satisfies the growth condition Gd（d 〉 1）, then generalized Weyl＇s theorem holds for f（T） for every f ∈ Hol（σ（T））. （v） If T ∈ SC, then, f（OssF＋ （T）） = aSBF＋ （f（T）） for all f ∈ Hol（σ（T））. （vi） Let T be a-isoloid such that T＊ ∈ HC. If T - AI has finite ascent at every A ∈ Eσ（T） and if F is of finite rank on Hsuch that TF = FT, then T ∈ F obeys generalized a-Weyl＇s theorem.

Hypercyclic和Supercyclic算子

本文给出了一个判别Ｈｙｐｅｒｃｙｃｌｉｃ （Ｓｕｐｅｒｃｙｃｌｉｃ）算子的新的方 法，推广了已文献中的结果。

HYPERCYCLIC OPERATORS ON QUASI-MAZUR SPACES

Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, KSthe （LF）-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.

一致Fredholm指标性质与(ω^1)性质

根据一致Fredholm指标性质定义了一种新的谱集,利用该谱集给出了Hilbert空间中有界线性算子满足(ω1)性质的充要条件.此外,研究了hypercyclic算子(或supercyclic算子)和(ω1)性质之间的关系,同时给出了hypercyclic算子与supercyclic算子新的判定方法.

Evolution paths of green economy modes and their trend of hypercycle economy

So far there is a lack of specialized research on the entire path and the law of the evolution of green economy modes,especially studies on its future trend.Some studies have been done only on its sub-process,in which there are some shortcomings such as overlapping and incomplete classification of economy modes,as well as the lack of mechanism models to describe the modes.This paper attempts to solve those problems.Firstly,various green economy modes are extracted and classified,and then their evolution paths are identified and reviewed,according to sustainable development theory,hypercycle theory,etc.,and to practical investigation.Secondly,theoretical models for each green economy mode and their multilevel dynamic evolution models in a practical field(forestry as an example)are established by the modeling method of system structure.Finally,the evolution law of green economy mode is analyzed systematically by means of the above-mentioned models.According to the research,some conclusions are drawn as follows:First,green economy modes have evolved through three stages as a whole.The first is the start-up stage,in which the mode is the‘from cradle to grave’end-treatment green economy(GE-I).The second is the growth stage,in which the mode is the‘from cradle to cradle’resources-chain closed-loop green economy(GE-Ⅱ).The third is the mature stage,in which the mode will be the‘from breeding to breeding‘three-chain hypercycle economy(GE-Ⅲ).Second,GEⅡhas evolved through a specific process.In depth,the basic framework of GEⅡis based on the circular economy theory;and then it is combined gradually with some other relevant theories and technologies that are cleaner production,low-carbon economy and sharing economy and so on.In breadth,GE-Ⅱis expanded layer by layer from unified enterprise GE-Ⅱto diversified enterprise(or cluster)GE-Ⅱ,and to societal GE-Ⅱ.The operational principle of GE-Ⅱcan be described by 3R theoretical model that includes 3 subsystems of reduction,recycle and reuse(3R),which are connected into an organic whole by resources chain.Third,GE-Ⅲis evolving through a specific process of three-chain(3C)stage by stage expansion,from resources-chain primary hypercycle to eco-chain secondary hypercycle,and to value-chain tertiary hypercycle.The operational principle of GE-Ⅲcan be described by 5R-3C theoretical model that includes 5 subsystems of reduction,recycle,reuse,recultivation and reallocation(5R),which are connected into an environment-economy self organizing system by resources chain,ecochain and value chain.In the system,resources chain is the foundation,eco-chain is the support,and value chain is the impetus.The 3C hypercycle promotes and catalyze each other to realize mutualism between ecosystem and industrial system.Fourth,the related field of GEⅢincludes the principal part of renewable-resource-based industry,and the expanded part of other relevant industries and even the whole society.Now in practice,GE-Ⅲhas initially formed.And in the future,it is the development tendency of green economy toward the establishment of the ecological civilization.

THE PROPERTY OF NONWANDERING OPERATOR

In this paper. we stady the nonwandering operator, which is a linear operator with chaos character and is in intnite dimensionol linear space. We give the hypercyclic bacomposition on the compact set of nonwandering operators.

Uniform Convergence of Translation Operators

We denote N, R, C the sets of natural, real and complex numbers respectively. Let (λn), n ∈ N be an unbounded sequence of complex numbers. Costakis has proved the following result. There exists an entire function f with the following property: for every x, y ∈ R with 0 , every θ ∈(0,1) and every a ∈ C there is a subsequence of natural numbers (mn), n ∈ N such that, for every compact subset L ⊆C , In the present paper we show that the constant function a cannot be replaced by any non-constant entire function G. This is so even if one demands the convergence in (*) only for a single radius r and a single positive number θ. This result is related with the problem of existence of common universal vectors for an uncountable family of sequences of translation operators.

Diskcyclicity of Sets of Operators and Applications

In this paper,we introduce and study the diskcyclicity and disk transitivity of a set of operators.We establish a diskcyclicity criterion and give the relationship between this criterion and the diskcyclicity.As applications,we study the diskcyclicty of Co-semigroups and C-regularized groups.We show that a diskcyclic Co-semigroup exists on a complex topological vector space X if and only if dim(X)=1 or dim(X)=∞and we prove that diskcyclicity and disk transitivity of C0-semigroups(resp C-regularized groups)are equivalent.

Left Multiplication Mappings on Operator Spaces

Let X be a separable infinite dimensional Banach space and B（X） denote its operator algebra, the algebra of all bounded linear operators T ： X → X. Define a left multiplication mapping LT ： B（X）→B（X） by LT（V） → TV, V ∈ B（X）. We investigate the connections between hypercyclic and chaotic behaviors of the left multiplication mapping LT on S（Z） and that of operator T on X. We obtain that LT is SOT-hypercyclic if and only if T satisfies the Hypercyclicity Criterion. If we define chaos on B（X） as SOT-hypercyclicity plus SOT-dense subset of periodic points, we also get that LT is chaotic if and only if T is chaotic in the sense of Devaney.

Property(ω)and the Single-valued Extension Property

By the new spectrum originated from the single-valued extension property,we give the necessary and sufficient conditions for a bounded linear operator defined on a Banach space for which property(ω)holds.Meanwhile,the relationship between hypercyclic property(or supercyclic property)and property(ω)is discussed.

On Boundary Factors and Traces of Subgroups of Finite Groups

A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.For any G-boundary factor H/AG of A,we call the subgroup(A∩H)/AG of G/AG a G-trace of A.On the basis of these notions,we give some new characterizations of hypercyclically embedded subgroups.

HYPERCYCLIC MULTIPLICATION COMPOSITION OPERATORS ON WEIGHTED BANACH SPACE

This paper characterizes some sufficient and necessary conditions for the hypercyclicity of multiples of composition operators on Hlog,0∞.