Invariant manifolds of hypercyclic vectors for the real scalar case
Juan P. Bes
证明了实局部凸空间上的超循环算子都存在一个稠密不变的超循环向量线性流形。
We show that every hypercyclic operator on a real locally convex space admits a dense, invariant linear manifold of hypercyclic vectors.
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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 …
Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a (bounded or unbounded) scalar type spectral operator $A$ in a complex Banach space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of the exponentials of such an operator, which, under a certain condition on the spectrum of the operator $A$, coincides with the $C_0$-semigroup generated by $A$. The spectrum of $A$ lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group $\left\{e^{tA}\right\}_{t\in {\mathbb R}}…
We investigate the hypercyclic properties of commutator maps acting on separable ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the space of compact operators on $\ell^2$. We also establish some necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.
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。
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。
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。
We provide criteria for the existence of upper frequently hypercyclic subspaces and for common hypercyclic subspaces, which include the following consequences. There exist frequently hypercyclic operators with upper-frequently hypercyclic subspaces and no frequently hypercyclic subspace. On the space of entire functions, each differentiation operator induced by a non-constant polynomial supports an upper frequently hypercyclic subspace, and the family of its non-zero scalar multiples has a common hypercyclic subspace. A question of Costakis and Sambarino on the existence of a common hypercycli…
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…
Universal holomorphic maps with slow growth II. functional analysis methods
借助超循环算子理论，本文证明了复欧几里得空间间具有慢增长率的超循环全纯映射的抽象丰富性，并证明了从$\mathbb{C}^n$到连通Oka流形$Y$的全纯映射空间中存在稠密的公共频繁超循环元素。
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…
Three results in linear dynamics
本文研究了线性动力学中的三个问题，证明了Fréchet空间不支持强超循环加权复合算子，计算了加权后移算子的常数以寻找强超循环标量倍数，并肯定回答了关于超循环向量的公开问题。
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…
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