Analytic automorphism group and similar representation of analytic functions

Abstract

In geometry group theory, one of the milestones is Gromov’s polynomial growth theorem: Finitely generated groups have polynomial growth if and only if they are virtually nilpotent. Inspired by Gromov’s work, we introduce the growth types of weighted Hardy spaces. In this paper, we focus on the weighted Hardy spaces of polynomial growth, which cover the classical Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and much broader. Our main results are as follows. [Formula: see text] We obtain the boundedness of the composition operators with symbols of analytic automorphisms of unit open disk acting on weighted Hardy spaces of polynomial growth, which implies the multiplication operator [Formula: see text] is similar to [Formula: see text] for any analytic automorphism [Formula: see text] on the unit open disk. Moreover, we obtain the boundedness of composition operators induced by analytic functions on the unit closed disk on weighted Hardy spaces of polynomial growth. [Formula: see text] For any Blaschke product [Formula: see text] of order [Formula: see text], [Formula: see text] is similar to [Formula: see text], which is an affirmative answer to a generalized version of a question proposed by Douglas in 2007. [Formula: see text] We also give counterexamples to show that the composition operators with symbols of analytic automorphisms of unit open disk acting on a weighted Hardy space of intermediate growth could be unbounded, which indicates the necessity of the setting of polynomial growth condition. Then, the collection of weighted Hardy spaces of polynomial growth is almost the largest class such that Douglas’s question has an affirmative answer. [Formula: see text] Finally, we give the Jordan representation theorem and similarity classification for the analytic functions on the unit closed disk as multiplication operators on a weighted Hardy space of polynomial growth.