Commutant lifting in several variables

TL;DR

This paper explores explicit commutant lifting results for weighted Bergman spaces over the unit ball and polydisc in several complex variables, extending classical one-variable results.

Contribution

It provides new explicit commutant lifting theorems for weighted Bergman spaces in multiple variables, including the unit ball and polydisc.

Findings

Explicit commutant lifting results for several variables

New results applicable even in the one-variable case

Extension of classical theorems to weighted Bergman spaces

Abstract

In this article we study commutant lifting, more generally intertwining lifting, for different reproducing kernel Hilbert spaces over two domains in $\mathbb{C}^n$, namely the unit ball and the unit polydisc. The reproducing kernel Hilbert spaces we consider are mainly weighted Bergman spaces. Our commutant lifting results are explicit in nature and that is why these results are new even in one variable $(n=1)$ set up.