Syndetically Hypercyclic Operators

Given a continuous linear operator T L(x) defined on a separable -space X, we will show that T satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers such that the sequence is hypercyclic. In contrast we will also prove that, for any hypercyclic vector x X of T, there exists a strictly increasing sequence such that and is somewhere dense, but not dense in X. That is, T and do not share the same hypercyclic vectors.     

On the Decomposition of the Riesz Operator and the Expansion of the Riesz Semigroup

For a Riesz operator T on a reflexive Banach space X with nonzero eigenvalues denote by E(λ_i; T) the eigen-projection corresponding to an eigenvalue λ_i. In this paper we will show that if the operator sequence is uniformly bounded, then the Riesz operator T can be decomposed into the sum of two operators T_p and T_r: T = T_p + T_r, where T_p is the weak limit of T_n and T_r is quasi-nilpotent. The result is used to obtain an expansion of a Riesz semigroup T(t) for t ≥ τ. As an application, we consider the solution of transport equation on a bounded convex body.     

广义复合二项风险模型的若干大偏差结果

This paper is a further investigation of large deviation for partial and random sums of random variables, where {Xn,n ≥ 1} is non-negative independent identically distributed random variables with a common heavy-tailed distribution function F on the real line R and finite mean μ∈ R. {N（n）,n ≥ 0} is a binomial process with a parameter p ∈ （0,1） and independent of {Xn,n ≥ 1}; {M（n）,n ≥ 0} is a Poisson process with intensity λ 〉 0, Sn = ΣNn i=1 Xi-cM（n）. Suppose F ∈ C, we futher extend and improve some large deviation results. These results can apply to certain problems in insurance and finance.     

Spectral theory and hypercyclic subspaces

A vector in a Hilbert space is called hypercyclic for a bounded operator if the orbit is dense in . Our main result states that if satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for . The converse is true even if is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.

     

Rotations of Hypercyclic and Supercyclic Operators

Let T B(X) be a hypercyclic operator and a complex number of modulus 1. Then T is hypercyclic and has the same set of hypercyclic vectors as T. A version of this results gives for a wide class of supercyclic operators that x X is supercyclic for T if and only if the set {tTⁿ x : t > 0, n = 0, 1, ...} is dense in X. This gives answers to several questions studied in the literature.     

Hypercyclic Conjugate Operators

We prove that for any weighted backward shift B = B_w on an infinite dimensional separable Hilbert space H whose weight sequence w = (w_n) satisfies , the conjugate operator is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an such that is dense in S(H). We generalize the result to more general conjugate maps , and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.     

Complete Convergence for Weighted Sums of Negatively Superadditive Dependent Random Variables

Let $\{X_n,n\geq1\}$ be a sequence of negatively superadditive dependent (NSD, in short) random variables and $\{a_{nk}, 1\leq k\leq n, n\geq1\}$ be an array of real numbers. Under some suitable conditions, we present some results on complete convergence for weighted sums $\sum_{k=1}^na_{nk}X_k$ of NSD random variables by using the Rosenthal type inequality. The results obtained in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.     

和的乘积的重对数律

设{X,X_n,n≥1}是独立的或φ -混合的或 ρ -混合的正的平稳随机变量序列，或$\{X,X_n,n≥1}$是正的随机变量序列使得{X_n-EX,n≥1\} 是平稳遍历的鞅差序列，记S_n=\sum\limitsⁿ_{j=1}X_j, n≥1 . 该文在条件EX=μ> 0 及0 Var(X)<∞下，证明了部分和的乘积$\prod\limits^n_{j=1}S_j/n!\mu^n$在合适的正则化因子下的某种重对数律.     

具误差隐格式迭代逼近严格伪压缩映像族公共不动点 (英)

设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设K是实Banach空间E中非空闭凸集，{Ti}i=1^N是N个具公共不动点集F的严格伪压缩映像，{αn}包括于[0,1]是实数例，{un}包括于K是序列，且满足下面条件（i）0〈α≤αn≤1;（ii）∑n=1∞（1-αn）=＋∞.（iii）∑n=1∞ ‖un‖〈＋∞.设x0∈K,{xn}由正式定义xn=αnxn-1＋（1-αn）Tnxn＋un-1,n≥1,其中Tn=Tnmodn，则下面结论（i）limn→∞‖xn-p‖存在，对所有p∈F；（ii）limn→∞d（xn,F）存在，当d（xn,F）=infp∈F‖xn-p‖；（iii）lim infn→∞‖xn-Tnxn‖=0.文中另一个结果是，如果{xn}包括于[1-2^-n,1]，则{xn}收敛，文中结果改进与扩展了Osilike（2004）最近的结果，证明方法也不同。     

Hypercyclic composition operators on Hv0‐spaces

We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator C_φ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators C_φ on $H_{v}^0$.     

Hypercyclic Pairs of Coanalytic Toeplitz Operators

A pair of commuting operators, (A,B), on a Hilbert space is said to be hypercyclic if there exists a vector such that {A ⁿ B ^k x : n, k ≥ 0} is dense in . If f, g ∈H ^∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M ^* _f , M ^* _g ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H ^∞(G) such that the pair (M ^* _f , M ^* _g ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Evolution systems and perturbations of generators of strongly continuous groups

Suppose A generates a strongly continuous linear group on a Banach space X and B is a linear operator on X. It is shown that an extension of generates a strongly continuous semigroup if and only if the family of operators has an appropriate evolution system. This produces simple sufficient conditions for an extension of to generate a strongly continuous semigroup, including

限制零的m进制数集之Hausdorff测度

对于任意整数表示mkz的分数部分.给出了数集Fm的Hausdorff测度是Fm的HausdorfF维数.     

随机环境下上临界分值过程的偏差不等式

令$\{Z_{n}, n\ge 0\}$为独立同分布随机环境下的上临界分支过程$\xi=(\xi_n)_{n\geq 0}$.本文给出了$\ln (Z_{n+n_{0}}/Z_{n_{0}})$的一些偏差不等式及其在构造置信区间上的一些应用.     

The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators

We continue here the line of investigation begun in [7], where we showed that on every Banach spaceX=l ₁⊗W (whereW is separable) there is an operatorT with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vectorx inX, the translatesT ^r x (r=1, 2, 3,...) are dense inX. This is an interesting result even if stated in a form which disregards the linearity ofT: it tells us that there is a continuous map ofX{0\{ into itself such that the orbit {T ^rx :r≧0{ of anyx teX \{0\{ is dense inX \{0\{. The methods used to construct the new operatorT are similar to those in [7], but we need to have somewhat greater complexity in order to obtain a hypercyclic operator.     

关于杨全会和陈永高提出的一个问题

对任意的正整数与集合,令为解的个数.杨全会和陈永高证明了:若整数且,则不存在集合使得对所有充分大的整数成立,其中.对整数和,定义为满足对所有整数成立的集合的个数.杨全会和陈永高证明了是有限的,且.同时,他们问对任意整数,是否存在使得对所有整数成立.在本文中,我们给出了在时的准确公式.从而推出在时成立.     

NA阵列加权乘积和的完全收敛性

设{X_(ni):1≤i≤n,n≥1}为行间NA阵列,g(x)是R~+上指数为α的正则变化函数,r>0,m为正整数,{a_(ni):1≤i≤n,n≥1}为满足条件(?)|a_(ni)|=O((g(n))~1)的实数阵列,本文得到了使sum from n=1 to ∞n~(r-1)Pr(|■multiply from j=1 to m a_(nij) X_(nij)|>ε)<∞,■ε>0成立的条件,推广并改进了Stout及王岳宝和苏淳等的结论。     

一类二元关系的公共后继指数集

设$V=\{ a_{1},a_{2},\ldots ,a_{n}\}$是$n\geq 2$的一个有限集合,$V$上所有本原的二元关系组成的集合记为$P_{n}(V)$.对任意的$Q\in P_{n}(V)$,与$Q$对应的有向图记为$G(Q)$.记$ P_{n}(V,d)=\{Q:Q\in P_{n}(V)$ 且$G(Q)$ 恰好包含 $d$ 个环\},其中$0相似文献   

Large Deviations for Proportions of Observations Which Fall in Random Sets Determined by Order Statistics

Let {X _n :n?≥?1} be independent random variables with common distribution function F and consider $K_{h:n}(D)=\sum_{j=1}^n1_{\{X_j-X_{h:n}\in D\}}$ , where h?∈?{1,...,n}, X _1:k ?≤???≤?X _k:k are the order statistics of the sample X ₁,...,X _k and D is some suitable Borel set of the real line. In this paper we prove that, if F is continuous and strictly increasing in the essential support of the distribution and if $\lim_{n\to\infty}\frac{h_n}{n}=\lambda$ for some λ?∈?[0,1], then $\{K_{h_n:n}(D)/n:n\geq 1\}$ satisfies the large deviation principle. As a by product we derive the large deviation principle for order statistics $\{X_{h_n:n}:n\geq 1\}$ . We also present results for the special case of Bernoulli distributed random variables with mean p?∈?(0,1), and we see that the large deviation principle holds only for p?≥?1/2. We discuss further almost sure convergence of $\{K_{h_n:n}(D)/n:n\geq 1\}$ and some related quantities.