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.
Related
- Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
- Compact and Hilbert-Schmidt composition operators on weighted Hardy spaces
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- Compactness of composition operators on Hardy spaces of homogeneous rooted trees
- Estimates of Weighted Hardy Operators and Commutators on Total Morrey Spaces
BibTeX
@misc{masuda2026weighted,
title = {{Weighted composition operators and classification of weighted Hardy spaces}},
author = {David Masuda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00105},
eprint = {2609.00105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00105},
}
RIS
TY - UNPB
TI - Weighted composition operators and classification of weighted Hardy spaces
AU - David Masuda
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00105
UR - https://arxiv.org/abs/2609.00105
AB - 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.
ER -