Characterization of weighted Hardy spaces on which all composition operators are bounded

TL;DR

This paper characterizes the weighted Hardy spaces where all composition operators are bounded, showing they must have weights that are essentially decreasing and slowly oscillating, with applications to automorphisms.

Contribution

It provides a complete characterization of weights for which all composition operators are bounded on weighted Hardy spaces, including conditions for automorphisms.

Findings

All composition operators are bounded iff weights are essentially decreasing and slowly oscillating.

Automorphisms induce bounded composition operators iff weights are slowly oscillating.

Applications demonstrating the implications of these characterizations.

Abstract

We give a complete characterization of the sequences $\beta = (\beta_n)$ of positive numbers for which all composition operators on $H^2 (\beta)$ are bounded, where $H^2 (\beta)$ is the space of analytic functions $f$ on the unit disk ${\mathbb D}$ such that $\sum_{n = 0}^\infty |a_n|^2 \beta_n < + \infty$ if $f (z) = \sum_{n = 0}^\infty a_n z^n$. We prove that all composition operators are bounded on $H^2 (\beta)$ if and only if $\beta$ is essentially decreasing and slowly oscillating. We also prove that every automorphism of the unit disk induces a bounded composition operator on $H^2 (\beta)$ if and only if $\beta$ is slowly oscillating. We give applications of our results.