Evgeny Abakumov, Julia Gordon
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.
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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.
Hypercyclicity of operators that $λ$-commute with the Hardy backward shift
本文研究满足BX=λXB的算子X的动力学性质，推广了Godefroy和Shapiro关于Hardy后移算子交换子中超循环元素的经典刻画。
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.
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…
We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.
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。
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。
We study the injective and surjective hull of operator ideals generated by hypercyclic backward weighted shifts that factor through $\ell^p$.
In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a question raised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on $\ell_1(\mathbb{N})$ with the convolution product for the family of backward shifts $(B_{w(λ)})_{λ>0}$ induced by the weights $w_n(λ)=1+λ/n$.
We characterize disjoint and simultaneously hypercyclic tuples of unilateral pseudo-shift operators on $\ell^p(\mathbb{N})$. As a consequence, complementing the results of Bernal and Jung, we give a characterization for simultaneously hypercyclic tuples of unilateral weighted shifts. We also give characterizations for unilateral pseudo-shifts that satisfy the Disjoint and Simultaneous Hypercyclicity Criterions. Contrary to the disjoint hypercyclicity case, tuples of weighted shifts turn out to be simultaneously hypercyclic if and only if they satisfy the Simultaneous Hypercyclicity Criterion.
It is not known if the inverse of a frequently hypercyclic bilateral weighted shift on $c_0(\mathbb{Z})$ is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.
Hypercyclic shifts on lattice graphs
本文将有向树上加权移位的超循环研究推广到非树图，通过格图等简单例子考察相应后向移位的超循环性。
We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $\mathcal U$-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on $c_0$. The construction of these shifts lies on the constr…
Bayart and Ruzsa [Ergodic Theory Dynam. Systems 35 (2015)] have recently shown that every frequently hypercyclic weighted shift on $\ell^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Proc. London Math. Soc. (3) 94 (2007)] who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. We first generalize the Bayart-Ruzsa theorem to all Banach sequence spaces in which the unit sequences are a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, …
Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I.$ In this article, we characterize disjoint hypercyclic and supercyclic weighted pseudo-shift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I.$ Also, the shifts on weighted $L^p$ spaces of a directed tree and the operator weighted shifts on $\ell^2(\mathbb{Z,\mathcal{K}})$ are investigated as special cases.
Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces $\ell^{2}(\mathbb Z)$ and $\ell^{2}(\mathbb N)$ to be subspace-transitive, consequently, we show that the Herrero question (D. A. Herrero. Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991)179-190) holds true even on a subspace of a Hilbert space, i.e. there exists an operator $T$ such that both $T$ and $T^*$ are subspace-hypercyclic operators for some subspaces. We display the conditions on the direct sum of two invertable bilateral forward weighte…
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。
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。
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.
Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is investigated. The results are achieved by establishing an analogy between frequent hypercyclicity for the translation semigroup and for weighted pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts in spaces of vanishing sequences. Frequent hypercylic translation semigroups in weighted $L^p$-spaces are also characterized.
Hypercyclic Generalized Shift Operators
本文研究Segal代数上移位算子的线性动力学性质，并刻画标准Hilbert模上超循环广义双边移位算子。
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…
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.
We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for a family of unilateral pseudo-shifts on $c_0(\mathbb{N})$ and $\ell^p(\mathbb{N})$, $1\le p <\infty$. As an application, we characterize disjoint frequently hypercyclic weighted shifts. We give analogous results for the weaker notions of disjoint upper frequent and reiterative hypercyclicity. Finally, we provide counterexamples showing that, although the frequent hypercyclicity, upper frequent hypercyclicity, and reiterative hypercyclicity coincide for weighted shifts on $…
we study the hypercyclic and chaotic properties of the time varying weighted backward shift operator $(Tx)(t)=w(t)x(t+a)$ in $L_p(0,\infty)(1\leq p<\infty)$ and $C_0[0,\infty)$. And we also analyse the spectral structure of the operators if the spaces are complex.
We give a quantitative interpretation of the Frequent Hypercyclicity Criterion. Actually we show that an operator which satisfies the Frequent Hypercyclicity Criterion is necessarily A-frequently hypercyclic, where A refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the "frequency" measured by the Frequent Hypercyclicity Criterion. Moreover we construct an example of unilateral weighted shift which is frequently hypercyclic but not A-frequently hypercyclic on a particular s…
A note on disjoint hypercyclicity for invertible bilateral pseudo-shifts on $\ell^{p}(\mathbb{Z})$
本文研究了在$\ell^{p}(\mathbb{Z})$上可逆双侧伪移位的不相交超循环性，给出了相关例子并回答了关于不相交反复超循环的开放问题。
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。
We first generalize the results of León and Müller [Studia Math. 175(1) 2006] on hypercyclic subspaces to sequences of operators on Fréchet spaces with a continuous norm. Then we study the particular case of iterates of an operator T and show a simple criterion for having no hypercyclic subspace. Finally we deduce from this criterion a characterization of weighted shifts with hypercyclic subspaces on the spaces lp or c0, on the space of entire functions and on certain Köthe sequence spaces. We also prove that if P is a non-constant polynomial and D is the differentiation operator on the space …
We analyze the hypercyclicity, chaoticity, and spectral structure of (bounded and unbounded) weighted backward shifts in a nonclassical sequence space, which the space $l_1$ of summable sequences is both isometrically isomorphic to and continuously and densely embedded into. Based on the weighted backward shifts, we further construct new bounded and unbounded linear hypercyclic and chaotic operators both in the nonclassical sequence space and the classical space $l_1$, including those that are hypercyclic but not chaotic.
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 $\ell^{2}(\mathbb Z)$…
Zero-one law of orbital limit points for weighted shifts
本文为加权移位算子存在非零轨道极限点即超循环的结果提供了新证明，将其推广至一般序列空间与解析函数空间，并进一步探讨了弱序列递归与弱序列超循环的性质。
Optimal growth of upper frequently hypercyclic functions for some weighted Taylor shifts
本文研究了单位圆盘解析函数空间上加权Taylor移位算子的超循环与$\mathcal{U}$-频繁超循环函数在$L^p$-平均意义下的最优增长，通过引入上频繁超循环性的中间概念统一了已有结果。
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。
On several dynamical properties of shifts acting on directed trees
本文研究了有向树上逆移位算子的F-transitivity和拓扑F-复发性，揭示了无根树上算子的复发性与超循环性的等价关系，并探讨了有根树中超循环性的条件。
Hypercyclicity of weighted shifts on weighted Bergman spaces
本文针对加权Bergman空间上的加权后移位算子，基于系数泛函范数估计分析其连续性与动力学性质，给出径向权情形下超循环移位的完整刻画，得到了新的系数乘子结果。
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。
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…
Dynamical Properties of Perturbed Weighted Shifts
本文建立加权移位对角扰动为超循环、混沌及频繁混沌的充分条件，获这类算子跟踪性质的完整解析刻画，构造了具跟踪性质的频繁混沌受扰加权移位例子。
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.
The Forward and Backward Shift on the Hardy Space of a Tree
本文研究树的 Hardy 空间与小 Hardy 空间上的前、后移位，系统刻画其有界性、范数、等距条件、谱及超循环性。
$\mathcal F$-Transitivity of Backward Shifts on Lattice Graphs
本文研究了有向格图上后向平移的$\mathcal F$-传递性，给出了有限宽度格的精确权重刻画及无限象限格的等价条件，部分解决了相关开放问题。
Dynamics of shift operators on non-metrizable sequence spaces
该文研究不可度量化的LF空间上Schauder基对应后移位算子的动力学性质，刻画了Köthe协层序列空间上加权广义后移位的相关动力学特征，并证明量子力学湮灭算子在缓增分布空间上具备多种强动力学性质。
We initiate the study of the forward and backward shifts on the Lipschitz space of a tree, $\mathcal L$, and on the little Lipshitz space of a tree, ${\mathcal L}_0$. We determine that the forward shift is bounded both on $\mathcal L$ and on ${\mathcal L}_0$ and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift, we determine when it is bounded on $\mathcal L$ and on ${\mathcal L}_0$, we find the norm when the tree is homogeneous, we calculate the spectrum for the case when the tree is homogeneous, and we determine, for a general tree, when it…
Let $B$, $I$ be the unweighted backward shift and the identity operator respectively on $l^{\infty}(\mathbb{N})$, the space of bounded sequences over the complex numbers endowed with the supremum norm. We prove that $I+λB$ is locally topologically transitive if and only if $|λ|>2$. This, shows that a classical result of Salas, which says that backward shift perturbations of the identity operator are always hypercyclic, or equivalently topologically transitive, on $l^p(\mathbb{N})$, $1\leq p<+\infty$, fails to hold for the notion of local topological transitivity on $l^{\infty}(\mathbb{N})$. We…
Weak disjointness of hypercyclic operators
本文研究超循环算子的弱不交性，通过建立类比于拓扑动力学中Weiss-Akin-Glasner定理的定理，特征化混合算子相对于Furstenberg族的弱不交性，并通过实例区分不同类别的超循环算子。
Dynamics of weighted backward shifts on certain analytic function spaces
本文引入两类解析函数空间，研究加权后移算子的有界性、相似性及动力学性质，证明其超循环子空间存在，并推广至算子的函数。
We study the dynamical behaviour of weighted shifts defined on sequence spaces of a directed tree. In particular, we characterize their boundedness as well as when they are hypercyclic, weakly mixing and mixing.
Dynamics of Weighted Backward Shifts on Cesàro Spaces of Rooted Trees
本文研究无叶局部有限根树上Cesàro空间加权后移算子的动力学，刻画其有界性、传递性、超循环性及混沌性，并给出反例说明各性质间关系。
A bounded linear operator $T$ acting on a Banach space $\B$ is called weakly hypercyclic if there exists $x\in \B$ such that the orbit ${T^n x: n=0,1,...}$ is weakly dense in $\B$ and $T$ is called weakly supercyclic if there is $x\in \B$ for which the projective orbit ${λT^n x: λ\in \C, n=0,1,...}$ is weakly dense in $\B$. If weak density is replaced by weak sequential density, then $T$ is said to be weakly sequentially hypercyclic or supercyclic respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. Th…
We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. - A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. - A typical upper-triangular operator is ergodic in the Gaussian sense, whereas a typical operator of the form "diagonal plus backward unilateral weighted shift" is ergodic but has only countably …
We show that the bilateral backward shift on $\ell^p(\mathbb{Z},ω)$ that has a projective orbit with a non-zero limit point is supercyclic. This phenomenon holds also for $Γ$-supercyclicity, which extends a result obtained for the first time by Chan and Seceleanu. Moreover, we show that if $K$ is a compact subset of $\ell^p(\mathbb{N},ω)$ such that its orbit under the unilateral backward shift $B$ on $\ell^p(\mathbb{N},ω)$ has a non-zero weak limit point, then $B$ is hypercyclic.
Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W \subseteq \ell^\infty$ arbitrarily complicated by choosing $W$ appropriately.
Dynamics of weighted shifts on $\ell^p$-sums and $c_0$-sums
本文推广加权移位，用算子取代权重，研究其在$\ell^p$-和与$c_0$-和上的超循环性、混合性、混沌性等动力学性质，并与原算子比较，解释线性动力学经典判据。
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。
We generalize the concept of coarse hypercyclicity, introduced by Feldman in \cite{Fe1}, to that of coarse topological transitivity on open cones. We show that a bounded linear operator acting on an infinite dimensional Banach space with a coarsely dense orbit on an open cone is hypercyclic and a coarsely topologically transitive (mixing) operator on an open cone is topologically transitive (mixing resp.). We also "localize" these concepts by introducing two new classes of operators called coarsely $J$-class and coarsely $D$-class operators and we establish some results that may make these cla…
Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fréchet space $X$ if and only if $X$ is non-isomorp…
On spaceability of shifts-like operators on $L^p$
本文证明了 $L^p$ 空间上类移位算子超循环向量集合的空间性，并给出了弱混合耗散复合算子的刻画。
Mixing unilateral backward shifts
本文在桶型和超桶型拓扑序列空间上刻画混合单边后移算子，推广了已有结果，并通过非平凡例子验证了结论的有效性。
A bounded operator on a real or complex separable infinite-dimensional Banach space $Z$ is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation $T$ of a standard Lebesgue probability space $(X,\mathcal {B},μ)$, there exists an $A$-invariant probability measure $ν$ on $H$ with full support such that the two dynamical systems $(X,\mathcal {B},μ;T)$ and $(H,\mathcal {B}_{H},ν;A)$ are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or …
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