Related papers: Operators commuting with the Volterra operator are…

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic,…

We prove that a semigroup generated by a finitely many truncated convolution operators on $L^p[0,1]$ with $1\leq p<\infty$ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Chan and Seceleanu have shown that if a weighted shift operator on $\ell^p(\mathbb{Z})$, $1\leq p<\infty$, admits an orbit with a non-zero limit point then it is hypercyclic. We present a new proof of this result that allows to extend it to…

There is no supercyclic power bounded operator of class $C_{1{\textstyle\cdot}}.$ There exist, however, weakly l-sequentially supercyclic unitary operators$.$ We show that if $T$ is a weakly l-sequentially supercyclic power bounded operator…

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 $x\in \B$…

This paper discusses the existence of a sufficient condition for an operator to be weakly hypercyclic. We establish a weak hypercyclicity criterion, and thereupon we can answer questions 5.3 and 5.8 posed by Chan and Sanders in 2004.…

In this paper, we study the so-called Bishop operators $T _ \alpha$ on $L ^ p ([0, 1])$, with $\alpha \in (0, 1)$ and $1 < p < + \infty$, from the point of view of linear dynamics. We show that they are never hypercyclic nor supercyclic,…

For every fixed $\epsilon$ $\in$ (0, 1), we construct an operator on the separable Hilbert space which is $\delta$-hypercyclic for all $\delta$ $\in$ ($\epsilon$, 1) and which is not $\delta$-hypercyclic for all $\delta$ $\in$ (0,…

We prove the spaceability of the set of hypercyclic vectors for {\em shifts-like operators}. Shift-like operators appear naturally as composition operators on $L^p(X)$, when the underlying space $X$ is dissipative. In the process of proving…

We construct a class of Fourier multipliers whose associated operators are weak (1,1) bounded but fail to be weak (p, p) bounded for any 1 < p \leq \infty. Moreover, we show that this result is sharp.

Let $2\leq p<\infty$ and $X$ be a complex infinite-dimensional Banach space. It is proved that if $X$ is $p$-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space…

The purpose of this paper is to characterize weak supercyclicity for Hilbert-space contractions, which is shown to be equivalent to characterizing weak supercyclicity for unitary operators$.$ This is naturally motivated by an open question…

Bounded linear operators on separable Banach spaces algebraically similar to the classical Volterra operator $V$ acting on $C[0,1]$ are characterized. From this characterization it follows that $V$ does not determine the topology of…

We prove that a continuous linear operator $T$ on a topological vector space $X$ with weak topology is mixing if and only if the dual operator $T'$ has no finite dimensional invariant subspaces. This result implies the characterization of…

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 H be a separable, infinite dimensional Hilbert space and let S be a countable subset of H. Then most positive operators on H have the property that every nonzero vector in the span of S is cyclic, in the sense that the set of operators…

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.

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter $\alpha$ and study their trajectory behaviors. We showed that for any…

We derive asymptotic information on the iterates of a Volterra convolution operator acting on L^p(0,1), subject to a mild smoothness condition on the kernel. In particular, an asymptotically equal sequence of rank 1 operators is obtained,…

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…