Three Classification Results In The Theory Of Weighted Hardy Spaces On The Ball

TL;DR

This paper characterizes when certain Hilbert function spaces on the complex ball are isometrically isomorphic under automorphisms and provides criteria for isomorphisms between weighted Hardy spaces on the disk.

Contribution

It introduces a classification of Hilbert spaces on the ball based on isometric properties and characterizes weighted Hardy space isomorphisms via composition operators.

Findings

Subsets of the ball induce isometric subspaces under automorphisms.

Pairs of weighted Hardy spaces are isomorphic if their weights satisfy a specific criterion.

Provides a unified framework for understanding isometries in Hardy spaces.

Abstract

We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic automorphism of the ball taking one to the other. We also characterize pairs of weighted Hardy spaces on the unit disk which are isomorphic via a composition operator by a simple criterion on their respective sequences of weights.