2022/09/08 by Bingzhe Hou, Chunlan Jiang, Hou, Bingzhe +1
Mathematics · #30J10 #46E20 #46J25 #47B33 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Operator Algebras (math.OA) #Primary 46J40 #Representation Theory (math.RT) #Secondary 47B35
paper · pdf · doi:10.48550/arxiv.2209.03852
openalex publication_date 2022/09/08 · openalex created_date 2022/09/15 · openalex updated_date 2026/07/28
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 Mz is similar to Mφ for any analytic automorphism φ 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, MB is similar to \bigoplus1m Mz, 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.