Hypercyclic operators are subspace hypercyclic

Abstract

In this short note, we prove that for a dense set A⊂X (X is a Banach space) there is a non-trivial closed subspace M⊂X such that A∩M is dense in M. We use this result to answer a question posed in Madore and Martínez-Avendaño (2011) [9]. In particular, we show that every hypercyclic operator is subspace-hypercyclic.