Some necessary and sufficient conditions for Hypercyclicity Criterion
B Yousefi, H Rezaei
给出了可分希尔伯特空间上算子满足超循环性准则的充要条件。
We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion.
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A quantitative interpretation of the frequent hypercyclicity criterion
我们给出了一个定量的解释频繁的超循环准则。实际上，我们证明了满足频繁超循环准则的算子必然是A-频繁超循环的，这里A表示比自然下密度更尖锐的加权密度.按照这个顺序，我们展示了不同尺度的加权密度，这些密度对量化由频繁高循环性标准测量的“频率”感兴趣。此外，我们构造了一个在特定尺度上频繁超循环但非A-频繁超循环的单侧加权移位的例子。
A Note on Weak Hypercyclicity and Linear Fractional Composition Operator
本文讨论了算子是弱超循环算子的一个充分条件的存在性。我们建立了一个弱超循环判据，并据此回答了Chan和Sanders在2004年提出的问题5.3和5.8。最后，我们证明了对于特定类型的复合算子，弱超循环性与超循环性是等价的。
Let $T$ be a so-called operator of Read's type on a (real or complex) separable Banach space, having no non-trivial invariant subset. We prove in this note that $T\oplus T$ is then hypercyclic, i.e. that $T$ satisfies the Hypercyclicity Criterion.
We provide a criterion for $\varepsilon$-hypercyclicity. Also, we extend the ideas of Badea, Grivaux, Müller and Bayart to construct $\varepsilon$-hypercyclic operators which are not hypercyclic in a wider class of separable Banach spaces, including several classical Banach spaces. For instance, our result can be applied to separable infinite dimensional $L^p$ spaces and $C(K)$ spaces.
A remark on the frequent hypercyclicity criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation
本文将$L^p(\R_+,μ)$上的平移$C_0$ -半群的一个结果推广到某些加权复合$C_0$ -半群上，证明了该结果是混沌的且满足Mangino和Peris的频繁超循环准则.当处理一阶线性偏微分算子的初值问题时，这样的$C_0$ -半群以一种自然的方式出现.我们将我们的结果应用到数学生物学中的线性冯福斯特-Lasota方程。还研究了索伯列夫空间上的加权复合$C_0$ -半群.
On the direct sum of two bounded linear operators and subspace-hypercyclicity
本文证明了如果两个算子的直和是子空间超循环的（满足子空间超循环准则），则两个算子都是子空间超循环的（满足子空间超循环准则）.此外，若算子$T$满足子空间超循环准则，则$T\oplus T$也满足子空间超循环准则.在一定条件下，若$T\oplus T$是超循环的，则$T$满足子空间-超循环准则，且子空间-超循环算子满足子空间-超循环准则，这给出了定理2.3的“子空间-超循环”类比. (in Hereditarily hypercyclic operators，J. Funct.肛门，167：94- 112，1999由P. Bés和A. Peris）。
A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity
本文研究连续映射的度量传递、混合和超循环，引入了δ-拓扑传递、δ-拓扑混合及统一从下界的δ-混合，并证明了UFB-δ-TM蕴含δ-TM蕴含δ-TT。在可分F空间中，提出了δ-超循环准则，并证明其蕴含δ-超循环，同时经典超循环准则对任意δ>0都蕴含δ准则。
We give a sufficient condition for two operators to be disjointly frequently hypercyclic. We apply this criterion to composition operators acting on $H(\mathbb D)$ or on the Hardy space $H^2(\mathbb D)$. We simplify a result on disjoint frequent hypercyclicity of pseudo shifts of a recent paper of Martin et al. and we exhibit two disjointly frequently hypercyclic weighted shifts.
Recurrence properties of hypercyclic operators
本研究提出$\mathcal{A}$-频繁超循环性新概念以推广三类超循环算子概念，给出相应的$\mathcal{A}$-频繁超循环准则并证明其可刻画加权移位算子的该性质，还探究了超循环算子相关集合的密度性质与reiteratively超循环算子新概念。
We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset $Λ$ of the complex plane for the family {$λ$T : $λ$ $\in$ $Λ$} to have a common frequently hypercyclic vector. In passing, this permits us to eas…
By strengthening one of the hypotheses of a well-known sufficient condition for the hypercyclicity of linear operators in Banach spaces, we arrive at a sufficient condition for linear chaos and reveal consequences of the latter for inverses, powers, multiples, and spectral properties. Extending the results, familiar for bounded linear operators, we also show that the hypercyclicity of unbounded linear operators subject to the sufficient condition for hypercyclicity is inherited by their bounded inverses, powers, and unimodular multiples and that necessary conditions for linear hypercyclicity s…
A hypercyclicity criterion for non-metrizable topological vector spaces
该研究提出非可度量化序列可分拓扑向量空间上算子序列超循环性的充分条件，并用两类无法通过F-空间比较原理推导超循环性的算子实例验证，解决了相关公开问题。
Tensor products of recurrent hypercyclic semigroups
研究了巴拿赫空间上满足超循环性判据、回归超循环性判据或混沌的强连续半群的张量积.
$ρ$-Frequently Hypercyclic Operators
引入$\ ho$-频繁超循环概念以细化频繁超循环，通过校准函数$\ ho$替换分母，给出判别准则并研究加权后向移位。
Strong mixing measures for linear operators and frequent hypercyclicity
我们构造了F-空间上满足频繁超循环准则的算子的强混合不变测度。对于数列空间上的单向后移，稍加修改就可以得到精确的不变测度。
Frequent, disjoint hypercyclicity and strong topological transitivity of generalized weighted shift operators on Hilbert C-modules
本文针对可分Hilbert空间上紧算子C-代数对应的标准Hilbert C-模，研究广义双边加权移位算子的频繁超循环准则、混沌、不相交超循环性及基于Furstenberg族的F-传递性等动力学性质。
Self-similar fractals and common hypercyclicity
研究推广了Costakis-Sambarino关于公共超循环向量的定理至多维情形，揭示了其对一类自相似分形的重要推论，并应用于由Hölder连续曲线参数化的反向移位乘积族。
Subspace hypercyclicity
希尔伯特空间上的有界线性算子T是子空间M的子空间-超循环算子，如果存在一个向量，其在T下的轨道在一个相对稠密的集合中与子空间相交.我们构造的例子表明，子空间超循环是有趣的，包括一个非平凡的子空间超循环运营商，是不是超循环。有一个Kitai样的标准，这意味着子空间超循环，虽然一个子空间超循环运营商的频谱必须相交的单位圆，并不是每个组件的频谱将这样做。我们表明，像超循环，子空间超循环是一个严格的无限维现象。此外，紧算子或亚正规算子永远不可能是子空间超循环的。
Given a locally compact group $G$ and $1\leq p<\infty$, a sufficient condition ensuring the \emph{disjoint hypercyclicity} of finitely many weighted translations on $L^p(G)$ was investigated in this paper.
Upper frequent hypercyclicity and related notions
推广了Bayart和Ruzsa的结果，得到了上频繁超循环算子的Birkhoff型刻画和相应的上频繁超循环性准则.作为一个应用程序，我们的特点上频繁超循环加权后向移位数列空间，这反过来又使我们能够拿出各种反例，在线性动力学，大大简单于以前在文献中获得的。更一般地说，我们引入上Furstenberg家族$\mathcal{A}$的概念，并表明我们的主要结果保持$\mathcal{A}$ -超循环算子关于这样的家庭。
$q$-Frequent hypercyclicity in spaces of operators
给出了可分Banach空间上算子代数上$C_{R,T}(S)=RST$形式的线性映射是$q$ -频繁超循环的条件.特别地，如果$R$是一个满足$q$ -频繁超循环准则的有界算子，则证明了在可分Hilbert空间$H$上的所有紧算子空间$\mathcal{K}(H)$和所有自伴算子的实拓扑向量空间$\mathcal{S}(H)$上，映射$C_{R}(S)$ = $RSR^*$是$q$ -频繁超循环的.进一步我们给出了$C_{R,T}$是Schatten von Neumann类$S_p(H)$上的$q$ -频繁超循环的条件.我们还刻画了$C_{M^*_φ,M_ψ}$在Hardy空间的迹类上的频繁超循环性，其中符号$M_φ$表示与$φ$相关的乘法算子.
In this paper, we study the hypercyclicity of forward and backward shifts on weighted $L^p$ spaces of a directed tree. In the forward case, only the trivial trees may support hypercyclic shifts, in which case the classical results of Salas apply. For the backward case, nontrivial trees may support hypercyclic shifts. We obtain necessary conditions and sufficient conditions for hypercyclicity of the backward shift and, in the case of a rooted tree on an unweighted space, we show that these conditions coincide.
In this short note, we answer a question of Martin and Sanders [Integr. Equ. Oper. Theory, 85 (2) (2016), 191-220] by showing the existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing and, therefore, fail to satisfy the Disjoint Hypercyclicity Criterion. We also show that given an operator $T$ such that $T \oplus T$ is frequently hypercyclic, the set of operators $S$ such that $T, S$ are disjoint frequently hypercyclic but fail to satisfy the Disjoint Hypercyclicity Criterion is SOT dense in the algebra of bounded linear operators.
Considering a family of upper frequently hypercyclic operators we care about the existence of vectors which are upper frequently hypercyclic for any operator of this family. We establish sufficient conditions for a family of operators to have these vectors called common upper frequently hypercyclic vectors. Using this result, we then give a construction of such vectors. Finally we derive some applications to families of weighted shifts.
A bounded linear operator $T$ on a Banach space $X$ is called hypercyclic if there exists a vector $x \in X$ such that $orb{(x,T)}$ is dense in $X$. The Hypercyclicity Criterion is a well-known sufficient condition for an operator to be hypercyclic. One open problem is whether there exists a space where the Hypercyclicity Criterion is also a necessary condition. For a number of reasons, the spaces with very-few operators are some natural candidates to be a positive answer to that problem. In this paper, we provide a theorem that establishes some relationships for operators in these spaces.
On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications
本文引入F_Γ-超循环算子与F-超循环标量集概念，给出识别判据，证明伪移位算子在无界Γ下满足性质，并刻画超循环标量集的充要条件，部分解答了2016年提出的开放问题。
Motivated by a question posed by Sophie Grivaux concerning the regularity of the orbits of frequently hypercylic operators, we show the following: for any operator $T$ on a separable metrizable and complete topological vector space $X$ which is both frequently hypercyclic and piecewise syndetic hypercyclic, the lower density and upper Banach density of the recurrence set $\{n\geq 1: T^n x\in U\}$ are different, for any hypercyclic vector $x\in X$ for $T$, and a certain collection of non-empty open sets $U\subseteq X$. As an immediate consequence we got a sufficient condition for a chaotic oper…
Hypercyclicity of Toeplitz operators with smooth symbols
本文研究了具有光滑符号的Toeplitz算子在Hardy空间上的超循环性，提出了必要和充分条件，并扩展了先前相关结果。
Hypercyclic and supercyclic linear operators on non-Archimedean vector spaces
本文的主要目的是发展非阿基米德值域上拓扑向量空间上线性算子的超循环性和超循环性理论。证明了有限维空间上不存在超循环算子。此外，我们还分别给出了超循环性的充分和必要条件.可分$F$ -空间上的线性算子的超循环性）。证明了拓扑向量空间$X$上的线性算子$T$是超循环的，如果它满足超循环的（或超循环的）. Supercyclic）准则。我们考虑了$c_0$上的后向移位，刻画了这类移位的超循环性和超循环性.最后，我们研究了算子$λI+μB$的超循环性，超循环性，其中$I$是恒等算子，$B$是后移算子.我们注意到，非阿基米德情形和真实情形之间有本质的区别。
Hypercyclicity and Lipschitz-free operators
本文提出目标性质准则，用于通过非线性限制判定线性动力系统的混合性，并证明Lipschitz自由算子的超循环性等价于原映射具有目标性质。
Dynamics of composition operators induced by odometers
研究了由 odometers 诱导的 composition 运算符的动力学特性，发现其超循环性等属性等价，但与加权反向位移不同。
Orbits of coanalytic Toeplitz operators and weak hypercyclicity
证明了Banach空间上有界线性算子弱超循环性的一个新判据。应用这个标准，我们解决了一些悬而未决的问题。也就是说，我们证明了，如果$G$是$\C$的一个区域，该区域由光滑Jordan曲线$Γ$限定，使得$G$不满足单位球，但$Γ$与单位圆相交于非平凡弧，则$M^*$是$H^2(G)$上的弱超循环算子，其中$M$是与自变量算子$Mf(z)=zf(z)$的乘积。证明了若$g$是单位圆盘$\D$上的Hardy空间$H^\infty(\D)$的非常数函数，使得$g(\D)\cap\D=\varnothing$和集合$\{z\in\C:|z|=1,\ |g(z)|=1\}$是正Lebesgue测度的单位圆$\T$的子集，则Hardy空间$H^2(\D)$上的共分析Toeplitz算子$T^*_g$是弱超循环的.相反，如果$g(\D)\cap\D=\varnothing$，$|g|>1$几乎处处在$\T$和$\log(|g|-1)\in L^1(\T)$上，则$T^*_g$不是1-弱超循环的，因此不是弱超循环的（复Banach空间$X$上的有界线性算子$T$称为$n$ -弱超循环的，如果存在$x\in X$使得对于每个满射连续线性算子$S:X\to \C^n$，集合$\{S(T^mx):m\in\N\}$在$\C^n$中稠密）。最后一个结果是基于较低的估计的成员的轨道的协分析Toeplitz运…
Disjoint Ces$\grave{a}$ro-hypercyclic operators
本文研究了不相交Ces$\grave{a}$ro-超循环算子，提出了相应的超循环准则并给出两种证明方法，同时详细刻画了序列空间上单边与双边加权移位算子的权序列特征。
Hypercyclic Toeplitz operators
研究了Hardy空间$H^2(\mathbb{D})$中Toeplitz算子的超循环性，其中$p(\bar{z}) +ϕ(z)$是多项式，$p$是多项式.我们发现的充要条件hypercyclicity几乎一致的情况下，当${\rm deg}\, p =1$。
Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity
该研究引入遗传频繁超循环性概念，证明其相关算子性质与构造反例，并解决不交频繁超循环性的多个问题，部分结果在Furstenberg族框架下得到。
Frequently hypercyclic semigroups
研究了强连续算子半群的频繁超循环性。更精确地说，我们给出了半群是频繁超循环的一个判据（充分条件），其形式依赖于Pettis积分。在某些情况下，这个判据可以用半群的无穷小生成元来证明。给出了Ornstein-Uhlenbeck算子生成的半群的应用，特别是$p$ -可积函数或连续函数的加权空间上的平移半群的应用.
We study hypercyclicity of Toeplitz operators in the Hardy space $H^2(\mathbb{D})$ with symbols of the form $R(\overline{z}) +ϕ(z)$, where $R$ is a rational function and $ϕ\in H^\infty(\mathbb{D})$. We relate this problem to cyclicity of certain families of functions for analytic Toeplitz operators and give new sufficient conditions for hypercyclicity based on deep results of B. Solomyak.
We show that, under suitable conditions, an operator acting like a shift on some sequence space has a frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. We also apply it to every operator satisfying the Frequent Hypercyclicity Criterion, recovering a result of Murillo and Peris.
On Topological Numerical Transitivity, C*-Transitivity and generalizations
本文在线性动力学中引入广义数值传递性与C*-传递性，证明二者与多种标准动力学概念不同，构造了相关算子并论证存在弱超循环但非C*-传递的算子。
Kitai's Criterion for Composition Operators
本文提出一个研究 composition 运算符动力学的框架，重新审视超循环和混合运算符的特征化，并展示满足 Kitai 准则的运算符与混合运算符的一致性。
q-Frequently hypercyclic operators
在局部凸空间上引入了q频繁超循环算子，并给出了连续算子是q频繁超循环算子的一个充分判据。给出了q-频繁超循环算子关于范数拓扑、F-范数拓扑和弱 *-拓扑的应用.最后，证明了由$T_μ(f)(z) = f'(μz)$，$μ\ge1$定义的非卷积算子$T_μ$在具有紧开拓扑的整函数空间$H(\mathbb{C})$上的频繁超循环性.
On Certain forms of Transitivities for Linear Operators
本文研究线性算子的多种传递性质，定义了基于Furstenberg家族的广义超循环准则，探讨了混合算子的等价特征，并揭示了混合算子与Kitai准则之间的差异。
A conditional weighted composition operator $T_u: L^p(Σ)\rightarrow L^p(\mathcal{A})$ ($1\leq p<\infty$), is defined by $T_u(f):= E^{\mathcal{A}}(u f\circ φ)$, where $φ: X\rightarrow X$ is a measurable transformation, $u$ is a weight function on $X$ and $E^{\mathcal{A}}$ is the conditional expectation operator with respect to $\mathcal{A}$. In this paper, we study the subspace-hypercyclicity of $T_u$ with respect to $L^p(\mathcal{A})$. First, we show that if $φ$ is a periodic nonsingular transformation, then $T_u$ is not $L^p(\mathcal{A})$-hypercyclic. The necessary conditions for the subspace…
Unbounded Chaotic Weighted Pseudo-shift
本文刻画了$\u2113^p$或$c_0$上无界超循环加权伪移位算子，并利用超循环判据给出了其成为混沌算子的充要条件。
Complementing earlier results on dynamics of unilateral weighted shifts, we obtain a sufficient (but not necessary, with supporting examples) condition for hypercyclicity, mixing and chaos for $M_z^*$, the adjoint of $M_z$, on vector-valued analytic reproducing kernel Hilbert spaces $\mathcal{H}$ in terms of the derivatives of kernel functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. Here $M_z$ denotes the multiplication operator by the coordinate function $z$, that is \[ (M_z f) (w) = w f(w), \] for all $f \in \mathcal{H}$ and $w \in \mathbb{D}$. We analyze the special case of quas…
Frequently hypercyclic $C_0$-semigroups indexed with complex sectors
本文研究了以复扇区为指标集的强连续算子半群的频繁超循环性，提出了修正定义，建立了充要条件，并导出了 $L^p_ρ(Δ, \mathbb{K})$ 空间上平移半群的实用判据与显式示例。
Dynamics of composition operators on spaces of holomorphic functions on plane domains
本文刻画了平面域上全纯函数空间中加权复合算子的超循环性与拓扑混合性之间的等价关系，并针对穿孔平面、穿孔圆盘及有限连通域等情形给出了超循环性的完整判据。
Some properties of subspaces-hypercyclic operators
本文肯定地回答了引入\cite{sub hyp}时提出的一个问题，即证明了：如果在Hilbert空间$\mathcal H$中$T$是具有$\mathcal M$ -超循环向量$x$的$\mathcal M$ -超循环算子，则$P(Orb(T,x))$在子空间$\mathcal M$中稠密，其中$P$是在$\mathcal M$上的正交投影.进一步，我们给出了${\mathcal M}^{\perp}$ -超循环性与在${\mathcal M}^{\perp}$上的正交投影之间的一些关系.给出了Hilbert空间$\ell^{2}(\mathbb Z)$上的双边加权移位算子是子空间超循环算子的充分条件，证明了存在算子$T$使得$T$和$T^*$都是子空间超循环算子.最后，我们给出了算子$T$的特征值的$\mathcal M$ -超循环准则.
Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree
本文研究了在根树上的小利普希茨空间中，满足频繁超循环准则的严格递增符号φ的特征，通过建立树上利普希茨型空间与经典空间ℓ^∞和c_0的自然同构，简化并改进了相关结果。
Hypercyclic and mixing composition operators on $\mathscr{O}_M(\mathbb{R})$
研究缓增光滑函数空间 $\mathscr{O}_M(\mathbb{R})$ 上的复合算子，刻画其混合性，结合 Abel 函数方程可解性给出序列超循环的充分条件，并据此证明许多混合复合算子具有超循环性。
On the dynamics of Toeplitz operators over Bergman spaces
本文研究 Bergman 空间上 Toeplitz 算子的超循环性，通过直接求解预解方程和绕数论证，在三对角情形下给出精确充要条件，消除了先前限制。
From weak to strong types of $L_E^1$-convergence by the Bocce-criterion
给出了弱收敛序列的振动性的充要条件（分别为相对弱紧集）在Bochner-Lebesgue空间$\l1$中是范数收敛的（分别为.相对范数紧），从而推广了$\rl1$的已知结果.同样地，给出了$\l1$从弱收敛到有限收敛（以及到Pettis范数收敛）的充分必要振动条件.证明了紧性是从有限收敛到强收敛的充要条件。在$\l1$的收敛模式之间的其他影响也进行了研究。
A compactness criterion in Schauder spaces
本文证明了 Schauder 空间中一个紧性的充分必要条件。
Criterion for convergence almost everywhere, with applications
本文给出了可测函数序列几乎必然收敛的充要条件，并考虑了它在傅里叶级数理论和随机场理论中的一些应用.
We provide necessary and sufficient conditions for the coincidence, up to equivalence of the norms, between strong and weak Orlicz spaces. Roughly speaking, this coincidence holds true only for the so-called exponential spaces. We find also the exact value of the embedding constant which appears in the corresponding norm inequality.
Consider a $C_0$-semigroup $(e^{tA})_{t \ge 0}$ on a function space or, more generally, on a Banach lattice $E$. We prove a sufficient criterion for the operators $e^{tA}$ to be positive for all sufficiently large times $t$, while the semigroup itself might not be positive. This complements recently established criteria for the individual orbits of the semigroup to become eventually positive for all positive initial values. We apply our main result to study the qualitative behaviour of the solutions to various partial differential equations.
Generalized Shemesh criterion, common invariant subspaces and irreducible completely positive superoperators
假设$A_{1},...,A_{s}$是复$n\times n$矩阵。本文给出了$A_{i}$存在公共特征向量的可计算判据，推广了D. Shemesh建立了两个矩阵。利用这一准则证明了当每个$A_{i}$有两两不同的特征值时，$A_{i}$有一个公共的$d$维不变子空间$2\leq d<n$的充要条件.最后，我们观察到所有具有多个特征值的矩阵的集合具有勒贝格测度0，因此该条件在实际应用中是充分的。受量子信息论的启发，我们给出了不可约完全正超算子的一些应用。
Hypercyclic Bergman-Toeplitz operators with some harmonic symbols
本文研究Bergman空间上符号为$a\ar{z}+p(z)$的Toeplitz算子的动力学性质，给出了超循环等性质的充要条件，并完整刻画了调和线性多项式符号的情形，还发现同一符号在不同空间上超循环行为可不同。
Recurrent subspaces in Banach spaces
本文研究Banach空间上算子的回复向量集的空间性，给出准刚性算子具有回复子空间的充分条件，并证明弱混合算子具有超循环子空间当且仅当它具有回复子空间。
Criterion for $\mathbb{Z}_d$--symmetry of a Spectrum of a Compact Operator
若$A$是Banach空间中的紧算子，且某个幂$A^q$是核的，则给出了$\mathbb{Z}_{d}$的一个判别准则--对所有充分大的$n$，$n \geq 0$，$n \neq 0 \mod d$，其谱$σ{A}$的迹$\mathop{\mathit{Trace}} A^n$消失对称.
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