Related papers: Invariant manifolds of hypercyclic vectors for the…

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…

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.

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 $|\lambda|\le 1$,…

Given a countable dense subset D of an infinite-dimensional separable Hilbert space H the set of operators for which every vector in D except zero is hypercyclic (weakly supercyclic) is residual for the strong (resp. weak) operator topology…

Let $(T\_\lambda)\_{\lambda\in\Lambda}$ be a family of operators acting on a $F$-space $X$, where the parameter space $\Lambda$ is a subset of $\mathbb R^d$. We give sufficient conditions on the family to yield the existence of a vector…

We treat some questions related to supercyclicity of continuous linear operators when acting in locally convex spaces. We extend results of Ansari and Bourdon and consider doubly power bounded operators in this general setting. Some…

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…

The sets of strongly supercyclic, weakly l-sequentially supercyclic, weakly sequentially supercyclic, and weakly supercyclic vectors for an arbitrary normed-space operator are all dense in the normed space, regardless the notion of…

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…

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 $\mathcal X$,…

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 show that the non-zero multiples of the derivative operator and the non-zero multiples of non-trivial translation operators on the space of entire functions share a common hypercyclic subspace, i.e. a closed infinite-dimensional subspace…

Let $\mathcal{H}$ be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator $T$ and its important relation with invariant subspace problem on $\mathcal{H}$: operator $T$ is…

It is proved that, if $(P_n)$ is a sequence of polynomials with complex coefficients having unbounded valences and tending to infinity at sufficiently many points, then there is an infinite dimensional closed subspace of entire functions,…

According to Grivaux, the group $GL(X)$ of invertible linear operators on a separable infinite dimensional Banach space $X$ acts transitively on the set $\Sigma(X)$ of countable dense linearly independent subsets of $X$. As a consequence,…

A subspace $X$ of a vector space over a field $K$ is hyperinvariant with respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is, every $…

We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.

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

We study the dynamics induced by an $m$-linear operator. We answer a question of B\`es and Conejero showing an example of an $m$-linear hypercyclic operator acting on a Banach space. Moreover we prove the existence of $m$-linear hypercyclic…

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…