Unilateral weighted shifts on ${\ell }^2$

Abstract

Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x\in {\ell }^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in ${\ell }^2$. We show that it is possible to make the set of $x\in {\ell }^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W\subseteq {\ell }^{\infty }$ arbitrarily complicated by choosing $W$ appropriately.

24 pages