Hypercyclicity in omega

Abstract:

A sequence $\mathbb {T}=(T_n)$ of operators $T_n :\mathscr {X}\to \mathscr {X}$ is said to be hypercyclic if there exists a vector $x\in \mathcal X$, called hypercyclic for $\mathbb {T}$, such that $\{ T_n x : n\geq 0 \}$ is dense. A hypercyclic subspace for $\mathbb {T}$ is a closed infinite-dimensional subspace of, except for zero, hypercyclic vectors. We prove that if $\mathbb {T}$ is a sequence of operators on $\omega$ that has a hypercyclic subspace, then there exist (i) a sequence $(p_n)$ of one variable polynomials $p_n$ such that $(p_n (\xi ))\in \omega$ is hypercyclic for every fixed $\xi$ and (ii) an operator $S:\omega \to \omega$ that maps nonzero vectors onto hypercyclic vectors for $\mathbb {T}$. We complement earlier work of several authors.