Hypercyclic sequences of operators
Volume 175 / 2006
Studia Mathematica 175 (2006), 1-18
MSC: Primary 47A16; Secondary 47A53.
DOI: 10.4064/sm175-1-1
Abstract
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.
