The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators

Abstract

We continue here the line of investigation begun in [7], where we showed that on every Banach spaceX=l ₁⊗W (whereW is separable) there is an operatorT with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vectorx inX, the translatesT ^r x (r=1, 2, 3,...) are dense inX. This is an interesting result even if stated in a form which disregards the linearity ofT: it tells us that there is a continuous map ofX{0\{ into itself such that the orbit {T ^rx :r≧0{ of anyx teX \{0\{ is dense inX \{0\{. The methods used to construct the new operatorT are similar to those in [7], but we need to have somewhat greater complexity in order to obtain a hypercyclic operator.