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

Abstract

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

arXiv admin note: substantial text overlap with arXiv:2011.14928