Weighted composition operators and classification of weighted Hardy spaces

Abstract

In this paper, we shed light on basic questions such as: What is the "right" underlying set on which a function space lives, and is this set uniquely determined? What are the appropriate morphisms to classify function spaces? When does a weighted composition operator induce an isomorphism, and when does it induce an isometric isomorphism? We first formulate suitable notions for domains of reflexive functional Banach spaces and study the corresponding notions of isomorphism between Hilbert function spaces. We then apply this framework to classify certain unitarily invariant spaces of holomorphic functions in several variables. Our main results extend Hartz's results from the complete Pick setting to the general setting of weighted Hardy spaces and establish a simple relation between the kernels of isomorphic spaces, improving also on earlier work of Ofek and Sofer in several directions. Additionally, we apply our results to investigate and estimate the Banach-Mazur distance of Hilbert function spaces.