A Reproducing Kernel Thesis for Operators on $\ell^2$--valued Bergman-type Function Spaces

TL;DR

This paper extends the reproducing kernel thesis to characterize boundedness and compactness of operators on $ ext{ell}^2$-valued Bergman-type spaces, generalizing classical results to vector-valued function spaces on various domains.

Contribution

It generalizes the reproducing kernel thesis for operators to $ ext{ell}^2$-valued Bergman-type spaces, including weighted spaces on multiple domains.

Findings

Characterization of boundedness and compactness via reproducing kernels

Extension of classical results to vector-valued Bergman spaces

Applicability to weighted spaces on various complex domains

Abstract

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on $\ell^2$--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their $\ell^2$--valued versions. In particular, the results in this paper apply to the weighted $\ell^2$--valued Bergman space on the unit ball, the unit polydisc and, more generally to weighted Fock spaces.