Hypercyclic sequences of operators

Fernando León-Saavedra¹ ; Vladimír Müller²

¹ Departamento de Matemáticas Facultad de Ciencias Universidad de Cádiz Pol. Rio San Pedro S/N 1500 Puerto Real, Spain
² Mathematical Institute Czech Academy of Sciences Žitná 25 115 67 Praha 1, Czech Republic

Studia Mathematica, Tome 175 (2006) no. 1, pp. 1-18

Résumé

A sequence $(T_n)$ of bounded linear operators between Banach spaces $X,Y$ is said to be hypercyclic if there exists a vector $x\in X$ such that the orbit $\{ T_nx\} $ is dense in $Y$. The paper gives a survey of various conditions that imply the hypercyclicity of $(T_n)$ and studies relations among them. The particular case of $X=Y$ and mutually commuting operators $T_n$ is analyzed. This includes the most interesting cases $(T^n)$ and $(\lambda _nT^n)$ where $T$ is a fixed operator and $\lambda _n$ are complex numbers. We also study when a sequence of operators has a large (either dense or closed infinite-dimensional) manifold consisting of hypercyclic vectors.

Détail

DOI : 10.4064/sm175-1-1

Keywords: sequence bounded linear operators between banach spaces said hypercyclic there exists vector orbit dense paper gives survey various conditions imply hypercyclicity studies relations among particular mutually commuting operators analyzed includes interesting cases lambda where fixed operator lambda complex numbers study sequence operators has large either dense closed infinite dimensional manifold consisting hypercyclic vectors