Related papers: A new class of frequently hypercyclic operators

A bounded linear operator $T$ on a Banach space $X$ is called frequently hypercyclic if there exists $x\in X$ such that the lower density of the set $\{n\in\N:T^nx\in U\}$ is positive for any non-empty open subset $U$ of $X$. Bayart and…

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…

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…

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…

According to Kim, Peris and Song, a continuous linear operator $T$ on a complex Banach space $X$ is called {\it numerically hypercyclic} if the numerical orbit $\{f(T^nx):n\in\N\}$ is dense in $\C$ for some $x\in X$ and $f\in X^*$…

The purpose of the present work is to treat a new notion related to linear dynamics, which can be viewed as a "localization" of the notion of hypercyclicity. In particular, let $T$ be a bounded linear operator acting on a Banach space $X$…

Even linear operators on infinite-dimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of…

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 $X$…

A sequence $\{T_n\}_{n=1}^{\infty}$ of bounded linear operators between separable Banach spaces $X, Y$ is called diskcyclic if there exists a vector $x\in X$ such that the disk-scaled orbit $\{\alpha T_n x: n\in \mathbb{N}, \alpha…

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe…

We study density properties of orbits for a hypercyclic operator $T$ on a separable Banach space $X$, and show that exactly one of the following four cases holds: (1) every vector in $X$ is asymptotic to zero with density one; (2) generic…

In this paper we first introduce the extended limit set $J_{\{T^n\}}(x)$ for a sequence of bounded linear operators $\{T_n\}_{n=1}^{\infty}$ on a separable Banach space $X$ . Then we study the dynamics of sequence of linear operators by…

We completely characterize the finite dimensional subsets A of any separable Hilbert space for which the notion of A-hypercyclicity coincides with the notion of hypercyclicity, where an operator T on a topological vector space X is said to…

We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular: we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace;…

We give an affirmative answer to a question asked by Faghih-Ahmadi and Hedayatian regarding supercyclic vectors. We show that if $\mathcal X$ is an infinite-dimensional normed linear space and $T$ is a supercyclic operator on $\mathcal X$,…

In this paper, we prove that if $T$ is diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of $T$ is dense in $\mathcal H$. Also, if $T$ is diskcyclic operator and $|\lambda|\le 1$,…

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if…

We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator \(T:X \to X\), we introduce the set \(\Omega(T)\), consisting of all continuous linear operators \(h:X \to X\) for…

Let $X$ be a real or complex Banach space and $T_t:X\to X$ is a power bounded operator (or a $C_0$-semigroup). If there exists a "occasionally" attracting compact subset K (for each x$ in unit ball $\liminf_n \rho(T^n x, K)=0$ then there…

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…