Dynamics of Weighted Backward Shifts on Certain Analytic Function Spaces

Abstract

We introduce the Banach spaces ?a,bp and c0,a,b, of analytic functions on the unit disc, having normalized Schauder bases consisting of polynomials of the form fn(z)=(an+bnz)zn,n?0, where {fn} is assumed to be equivalent to the standard basis in ?p and c0, respectively. We study the weighted backward shift operator Bw on these spaces, and obtain necessary and sufficient conditions for Bw to be bounded, and prove that, under some mild assumptions on {an} and {bn}, the operator Bw is similar to a compact perturbation of a weighted backward shift on the sequence spaces ?p or c0. Further, we study the hypercyclicity, mixing, and chaos of Bw, and establish the existence of hypercyclic subspaces for Bw by computing its essential spectrum. Similar results are obtained for a function of Bw on ?a,bp and c0,a,b. � The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.