Linear structure of hypercyclic vectors

Abstract

A vectorxin a Banach space B is called hypercyclic for a bounded linear operatorT:B→B if the orbit {Tnx:n≥1} is dense in B. Our main result states that ifTis a compact perturbation of an operator of norm ≤1 and satisfies an appropiate extra hypothesis, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors forT. In particular the result applies to compact perturbations of the identity. We also include applications to some weighted backward shifts and compact perturbations of the identity by weighted backward shifts. This last result in combination with a recent one that states that every Banach space admits an operator with a hypercyclic vector proves that in all Banach space there is an operatorTwith an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The main result also applies to the differentiation operator and the translation operatorT:f(z)→f(z+1) on certain Hilbert spaces consisting of entire functions. © 1997 Academic Press.