Compositional frequent hypercyclicity on weighted Dirichlet spaces

Abstract

It is proved that, in most cases, a scalar multiple of a linear-fractional generated composition operator $\lambda C_\varphi$ acting on a weighted Dirichlet space $S_\nu$ of holomorphic functions in the open unit disk is frequently hypercyclic if and only if it is hypercyclic. In fact, this holds for all triples $(\nu ,\lambda , \varphi )$ with the possible exception of those satisfying $\nu \in [1/4,1/2), \, |\lambda | = 1, \, \varphi =$ a parabolic automorphism.