Related papers: ${\mathcal F}$-Hypercyclic operators on Fr\' echet…

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…

Although recurrence for dynamical systems has been studied since the end of the nineteenth century, the study of recurrence for linear operators started with papers by Costakis, Manoussos and Parissis in 2012 and 2014. We explore recurrence…

In this paper, we define and study subspace-diskcyclic operators. We show that subspace-diskcyclicity does not imply to diskcyclicity. We establish a subspace-diskcyclic criterion and use it to find a subspace-diskcyclic operator that is…

We study the rate of growth of entire functions that are frequently hypercyclic with respect to some upper weighted densities for the differentiation operator. The statements obtained show the link between the minimal growth of frequently…

We answer one of the main current questions in Linear Dynamics by constructing a chaotic operator on $\ell^1$ which is not $\mathcal{U}$-frequently hypercyclic and thus not frequently hypercyclic. This operator also gives us an example of a…

In this paper we study some dynamical properties such as Frequent Hypercyclicity Criterion, chaos, disjoint hypercyclicity and F-transitivity via Furstenberg family F for generalized bilateral weighted shift operator on the standard Hilbert…

We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms…

We study shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is actually a closed…

We show that there exists an invertible $\mathcal{U}$-frequently hypercyclic operator on $\ell^p(\mathbb{N})$ ($1\le p <\infty$) whose inverse is not $\mathcal{U}$-frequently hypercyclic.

A Fr\'echet space $X$ satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace $M$ in $X$, each continuous operator on $M$ possesses a non-trivial invariant subspace (resp.…

We study dynamical notions lying between $\mathcal{U}$-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies…

Let $f:X\to X$ be an invertible Lipschitz transformation on a compact metric space $X$. Given a H\"{o}lder continuous invertible operator cocycles on a Banach space and an $f$-invariant ergodic measure, this paper establishes the H\"{o}lder…

We introduce and discuss Fr\'echet differentiability for maps between Fr\'echet spaces. For delay differential equations $x'(t)=f(x_t)$ we construct a continuous semiflow of continuously differentiable solution operators $x_0\mapsto x_t$,…

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…

We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fr\'{e}chet space of the entire mappings that are bounded on bounded sets the composition turns to be…

Bounded weighted composition operators, as well as compact weighted composition operators, on Fock spaces have been characterised. This characterisation is refined to the extent that the question of whether weighted composition operators on…

Let $B$ be an unilateral weighted backward shift on $\ell_p$, $1 \leq p < \infty$, that admits a $\mathscr{U}$-frequently hypercyclic subspace. We prove that $B$ admits such a subspace free of frequently hypercyclic vectors. The proof…

In this paper, we characterize supercyclic weighted composition operators on a large class of solid Banach function spaces, in particular on Lebesgue, Orlicz and Morrey spaces. Also, we characterize supercyclic weighted composition…

We introduce the super-shadowing property in linear dynamics, where pseudotrajectories are approximated by sequences of the form $(\lambda_nT^nx)$, with $(\lambda_n)_n$ being complex scalars. For compact operators on Banach spaces, we…

A weighted space of entire functions rapidly decreasing on the real line is considered in the paper. A growth of these functions along the imaginary axis is controlled by some system of weight functions. The Fourier transform of functions…