Existence of hypercyclic subspaces for Toeplitz operators

Abstract

In this work we construct a class of coanalytic Toeplitz operators, which have an infinite-dimensional closed subspace, where any non-zero vector is hypercyclic. Namely, if for a function $\varphi$ which is analytic in the open unit disc $\mathbb D$ and continuous in its closure the conditions $\varphi(\mathbb T)\cap\mathbb T\ne\emptyset$ and $\varphi(\mathbb D)\cap\mathbb T\ne\emptyset$ are satisfied, then the operator $\varphi(S^*)$ (where $S^*$ is the backward shift operator in the Hardy space) has the required property. The proof is based on an application of a theorem by Gonzalez, Leon-Saavedra and Montes-Rodriguez.