Analytic automorphism group and similar representation of analytic functions

TL;DR

This paper explores the properties of weighted Hardy spaces of polynomial growth, establishing boundedness of certain operators, classifying their similarities, and providing counterexamples to highlight the importance of growth conditions.

Contribution

It introduces the growth types of weighted Hardy spaces, proves boundedness of composition operators, and classifies similarity of multiplication operators on these spaces.

Findings

Boundedness of composition operators with automorphism symbols on polynomial growth spaces.

Similarity of multiplication operators induced by Blaschke products to direct sums of $M_z$.

Counterexamples showing unbounded composition operators on intermediate growth spaces.

Abstract

In geometry group theory, one of the milestones is M. Gromov's polynomial growth theorem: Finitely generated groups have polynomial growth if and only if they are virtually nilpotent. Inspired by M. 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. $(1)$ 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 $M_z$ is similar to $M_{\varphi}$ for any analytic automorphism $\varphi$ 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. $(2)$ For any Blaschke product $B$ of order $m$, $M_B$ is similar to $\bigoplus_{1}^m M_z$, which is an affirmative answer to a generalized version of a question proposed by R. Douglas in 2007. $(3)$ 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. $(4)$ 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.