Common hypercyclic vectors for composition operators

Luh超循环向量与复合算子

Luh hypercyclic vector for composition operator

本文构造单连通域上全纯函数，使其在复合算子下所有导数与反导数轨道稠密，证明Luh超循环向量构成稠密线性流形，并研究加权复合算子生成的余弦算子函数的超循环性条件。

Hypercyclic algebras

We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.

A Note on Weak Hypercyclicity and Linear Fractional Composition Operator

This paper discusses the existence of a sufficient condition for an operator to be weakly hypercyclic. We establish a weak hypercyclicity criterion, and thereupon we can answer questions 5.3 and 5.8 posed by Chan and Sanders in 2004. Lastly, we show that for specific type of composition operator, weak hypercyclicity and hypercyclicity are equivalent.

Invariant manifolds of hypercyclic vectors for the real scalar case

We show that every hypercyclic operator on a real locally convex space admits a dense, invariant linear manifold of hypercyclic vectors.

Common hypercyclic vectors for multiples of backward shift

We prove that the space $l^2$ contains a dense set of vectors which are hypercyclic simultaneously for all multiples of the backward shift operator by constants of absolute value greater than 1.

Common hypercyclic vectors and dimension of the parameter set

We investigate the existence of a common hypercyclic vector for a family $(T_λ)_{λ\in Λ}$ of hypercyclicoperators acting on the same Banach space $X$. We give positive and negative results involving the dimension of $Λ$ and the regularity of each map $λ\in Λ\mapsto T_λ^n x$, $x\in X$, $n\in\mathbb N$.

Common hypercyclic vectors for families of backward shift operators

We provide necessary and sufficient conditions on the existence of common hypercyclic vectors for multiples of the backward shift operator along sparse powers. Our main result strongly generalizes corresponding results which concern the full orbit of the backward shift. Some of our results are valid in a more general context, in the sense that they apply for a wide class of hypercyclic operators.

Frequently hypercyclic random vectors

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.

常见频繁超循环随机向量

Common frequently hypercyclic random vectors

研究可数家族加权逆向移位在ℓ_p空间中的共同频繁超循环向量，利用概率技术提出存在性准则并补充不存在性结果，应用于多项式加权逆向移位家族，提供共同频繁超循环向量的条件。

Algebrable sets of hypercyclic vectors for convolution operators

We show that several convolution operators on the space of entire functions, such as the MacLane operator, support a dense hypercyclic algebra that is not finitely generated. Birkhoff's operator also has this property on the space of complex-valued smooth functions on the real line.

Hypercyclicity of composition operators in Stein manifolds

We characterise hypercyclic composition operators $C_φ:f\mapsto f\circφ$ on the space of functions holomorphic on $Ω$, where $Ω$ is a connected Stein manifold and $φ$ is a holomorphic self-mapping of $Ω$. In the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in $Ω$, we show that much simpler characterisation is possible (many natural classes of domains in $\CC^N$ satisfy this condition). Moreover, we show that in such a class of manifolds, and in simply connected and infinitely connected planar domains, hypercyclicity of $C_φ$ implies its hereditary …

Hypercyclic operators on topological vector spaces

Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fréchet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.

Common frequent hypercyclicity

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…

Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator $T$ on a complex Fréchet space $X$ and a set $Λ\subseteq \R_+\times\C$ which is not of zero three-dimensional Lebesgue measure, the family $\{aT+bI:(a,b)\inΛ\}$ has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. I…

On the set of hypercyclic vectors for the differentiation operator

Let $D$ be the differentiation operator $Df=f'$ acting on the Fréchet space $\H$ of all entire functions in one variable with the standard (compact-open) topology. It is known since 1950's that the set $H(D)$ of hypercyclic vectors for the operator $D$ is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepúlveda whether the set $H(D)$ contains (up to the zero function) a non-trivial subalgebra of $\H$ or an infinite dimensional closed linear subspace of $\H$. In the present article both questions are answered affirmatively.

关于超循环空间和（共同）$\mathscr{U}$-频繁超循环空间

On hypercyclic and (common) upper frequently hypercyclic subspaces

本文研究了 unilateral weighted backward shift 在 $\ell_p$ 上的超循环空间性质，证明了存在不包含频繁超循环向量的超循环子空间，并解决了 Bès 和 Menet 2015 年提出的问题。

Frequent hypercyclicity and piecewise syndetic recurrence sets

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…

A class of Toeplitz operators with hypercyclic subspaces

We use a theorem by Gonzalez, Leon-Saavedra and Montes-Rodriguez to construct a class of coanalytic Toeplitz operators which have an infinite-dimensional closed subspace, where any non-zero vector is hypercyclic.

Banach空间上超循环算子轨道的密度性质

Density properties of orbits for a hypercyclic operator on a Banach space

研究Banach空间上超循环算子轨道的密度性质，证明四种情况中仅有一种成立，并通过例子展示所有情况均可能出现。

Hypercyclic Algebras for convolution operators of unimodular constant term

We study the existence of hypercyclic algebras for convolution operators $Φ(D)$ on the space of entire functions whose symbol $Φ$ has unimodular constant term. In particular, we provide new eigenvalue criteria for the existence of densely strongly algebrable sets of hypercyclic vectors.

Some Results on Subspace-Hypercyclic Operators

A bounded linear operator $T$ on a Banach space $X$ is called subspace-hypercyclic if there is a subspace $M \subsetneq X$ and a vector $x \in X$ such that $orb{(x,T)} \cap M$ is dense in $M$. We show that every Banach space supports subspace-hypercyclic operators and provide a new criteira for subspace-hypercyclic operators, generalizing a previous result from Le.

频繁超循环复合算子在根树上的小利普希茨空间

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。

Common upper frequent hypercyclicity

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.

Hypercyclic and mixing operator semigroups

We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group ${T_t}_{t\in\C^n}$ with holomorphic dependence on the parameter $t$. This result covers those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups ${T_t}_{t\geq 0}$.

Hypercyclic and supercyclic linear operators on non-Archimedean vector spaces

A main objective of the present paper is to develop the theory of hypercyclicity and supercyclicity of linear operators on topological vector space over non-Archimedean valued fields. We show that there does not exist any hypercyclic operator on finite dimensional spaces. Moreover, we give sufficient and necessary conditions of hypercyclicity (resp. supercyclicity) of linear operators on separable $F$-spaces. It is proven that a linear operator $T$ on topological vector space $X$ is hypercyclic (supercyclic) if it satisfies Hypercyclic (resp. Supercyclic) Criterion. We consider backward shifts…

Subspace-hypercyclic conditional type operators on $L^p$-spaces

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…

Hypercyclic behavior of some non-convolution operators on $H(\mathbb{C}^N)$

We study hypercyclicity properties of a family of non-convolution operators defined on spaces of holomorphic functions on $\mathbb{C}^N$. These operators are a composition of a differentiation operator and an affine composition operator, and are analogues of operators studied by Aron and Markose on $H(\mathbb{C})$. The hypercyclic behavior is more involved than in the one dimensional case, and depends on several parameters involved.

超超循环性及其与混合性质的联系

Ultra hypercyclicity and its connection to mixing properties

本文引入算子的超超循环性新拓扑性质，对比其与多种超循环、混合类性质的关系，证明相关算子性质结论，刻画特定空间上的超超循环加权后移位并改进已有必要条件。

Baire theorem and hypercyclic algebras

The question of whether a hypercyclic operator $T$ acting on a Fr{é}chet algebra $X$ admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterizations in the context of convolution operators acting on $H(\mathbb C)$ and backward shifts acting on a general Fr{é}chet sequence algebra.Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighte…

Relationships between Cyclic and Hypercyclic Operators

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.

Some properties of subspaces-hypercyclic operators

In this paper, we answer a question posed in the introduction of \cite{sub hyp} positively, i.e, we show that if $T$ is $\mathcal M$-hypercyclic operator with $\mathcal M$-hypercyclic vector $x$ in a Hilbert space $\mathcal H$, then $P(Orb(T,x))$ is dense in the subspace $\mathcal M$ where $P$ is the orthogonal projection onto $\mathcal M$. Furthermore, we give some relations between ${\mathcal M}^{\perp}$-hypercyclicity and the orthogonal projection onto ${\mathcal M}^{\perp}$. We also give sufficient conditions for a bilateral weighted shift operators on a Hilbert space $\ell^{2}(\mathbb Z)$…

Spectrum of a weakly hypercyclic operator meets the unit circle

It is shown that every component of the spectrum of a weakly hypercyclic operator meets the unit circle. The proof is based on the lemma that a sequence of vectors in a Banach space whose norms grow at geometrical rate doesn't have zero in its weak closure.

Linear Factorization of Hypercyclic Functions for Differential Operators

On the Fréchet space of entire functions $H(\mathbb{C})$, we show that every nonscalar continuous linear operator $L:H(\mathbb{C})\to H(\mathbb{C})$ which commutes with differentiation has a hypercyclic vector $f(z)$ in the form of the infinite product of linear polynomials: \[ f(z) = \prod_{j=1}^\infty \, \left( 1-\frac{z}{a_j}\right), \] where each $a_j$ is a nonzero complex number.

A new class of frequently hypercyclic operators

We study a hypercyclicity property of linear dynamical systems: a bounded linear operator T acting on a separable infinite-dimensional Banach space X is said to be hypercyclic if there exists a vector x in X such that {T^{n}x : n>0} is dense in X, and frequently hypercyclic if there exists x in X such that for any non empty open subset U of X, the set {n>0 ; T^n x \in U} has positive lower density. We prove that if T is a bounded operator on X which has "sufficiently many" eigenvectors associated to eigenvalues of modulus 1 in the sense that these eigenvectors are perfectly spanning, then T is…

Two Families of Hypercyclic Non-Convolution Operators

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $λ,b\in\mathbb{C}$, let $C_{λ,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{λ,b} f(z)=f(λz+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{λ,b}=C_{λ,b} \circ D$ by showing that whenever $|λ|\geq 1$, the collection of operators \begin{align*} \{ψ(T_{λ,b}): ψ(z)\in H(\mathbb{C}), ψ(0)=0 \text{ and } ψ(T_{λ,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual …

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is rel…

$q$-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form $C_{R,T}(S)=RST$ to be $q$-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if $R$ is a bounded operator satisfying the $q$-Frequent Hypercyclicity Criterion, then the map $C_{R}(S)$=$RSR^*$ is shown to be $q$-frequently hypercyclic on the space $\mathcal{K}(H)$ of all compact operators and the real topological vector space $\mathcal{S}(H)$ of all self-adjoint operators on a separable Hilbert space $H$. Further we provide a condition for $C_{R,T}$ to be $q$-frequently hypercyclic on the Schatten von Ne…

Common hypercyclic vectors for certain families of differential operators

Let (k(n)) n=1,2,... be a strictly increasing sequence of positive integers . We consider a specific sequence of differential operators Tk(n),λ , n=1,2,... on the space of entire functions , that depend on the sequence (k(n)) n=1,2,... and the non-zero complex number λ . We establish the existence of an entire function f , such that for every positive number λ the set {Tk(n),λ ,n=1,2,... } is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane . This provides the best possible strenghthened version of a corresponding r…

非可度量化拓扑向量空间的超循环性准则

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。

Banach 空间上超循环算子轨道的三分法

Trichotomy for the orbits of a hypercyclic operator on a Banach space

本文在可分 Banach 空间上建立了超循环算子轨道的三分法，区分均值渐近于零、绝对均值不规则和均值发散到无穷三种情形，并用加权后移算子说明三种情况均可实现。

Kitai 对 composition 运算符的准则

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。

Multiplicative structures of hypercyclic functions for convolution operators

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains.

由 odometers 引起的 composition 运算符的动力学

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。

Dynamics of weighted composition operators on function spaces defined by local properties

We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreo…

Common hypercyclic vectors for high dimensional families of operators

Let $(T\_λ)\_{λ\inΛ}$ be a family of operators acting on a $F$-space $X$, where the parameter space $Λ$ is a subset of $\mathbb R^d$. We give sufficient conditions on the family to yield the existence of a vector $x\in X$ such that, for any $λ\inΛ$, the set $\big\{T\_λ^n x;\ n\geq 1\big\}$ is dense in $X$. We obtain results valid for any value of $d\geq 1$ whereas the previously known results where restricted to $d=1$. Our methods also shed new light on the one-dimensional case.

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T; that there exist frequently hypercyclic operators on the sequence space c_0 admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vect…

Subspace hypercyclicity

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity is interesting, including a nontrivial subspace-hypercyclic operator that is not hypercyclic. There is a Kitai-like criterion that implies subspace-hypercyclicity and although the spectrum of a subspace-hypercyclic operator must intersect the unit circle, not every component of the spectrum will do so. We show that, like hypercyclicity, subspace-hypercyclic…

关于一类λ-次正规算子

On a class of lambda-hyponormal operators

本文定义了无限维希尔伯特空间上的λ-次正规算子，证明了一类此类算子不能超循环，并研究了加权复合算子的值域闭性与λ-次正规性，给出了相关例子。

树上加权移位算子的超循环代数

Hypercyclic algebras for weighted shifts on trees

本文研究有向树序列空间上加权后向移位算子的超循环向量代数存在性，给出 $c_0(V)$、$\ell^1(V)$ 上的代数结构结论及 $\ell^p(V)$ 上的充要条件，并探讨了混合性及无根树情形。

平面域上全纯函数空间中复合算子的动力学

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。

Mixing operators on spaces with weak topology

We prove that a continuous linear operator $T$ on a topological vector space $X$ with weak topology is mixing if and only if the dual operator $T'$ has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space $ω$ due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator $T$ on $ω$, $T\oplus T$ is also hypercyclic.

Frequently hypercyclic operators with irregularly visiting orbits

We prove that a bounded operator $T$ on a separable Banach space $X$ satisfying a strong form of the Frequent Hypercyclicity Criterion (which implies in particular that the operator is universal in the sense of Glasner and Weiss) admits frequently hypercyclic vectors with irregularly visiting orbits, i.e. vectors $x\in X$ such that the set $\mathcal{N}_T(x,U)=\{n\ge 1\,;\,T^{n}x\in U\}$ of return times of $x$ into $U$ under the action of $T$ has positive lower density for every non-empty open set $U\subseteq X$, but there exists a non-empty open set $U_0\subseteq X$ such that $\nt{x}{U_0}$ has…

扩展不交超循环算子族

Extending Families of Disjoint Hypercyclic Operators

本文解决 Salas 关于不交超循环算子的两个公开问题：证明任意不交超循环算子族都可在强算子拓扑稠密意义下扩展，并构造满足不交膨胀/坍缩性质却无稠密 d-超循环流形的不交弱混合算子。

关于复合算子的动力学：超循环性、里程计和平移

On the dynamics of composition operators: supercyclicity, odometers and translations

本文研究可测函数Banach空间上复合算子的动力学性质，特别通过里程计诱导的复合算子给出新例子和反例，并得到L_p空间上复合算子的超循环性和频繁超循环性的一般结论。