# Commutant lifting and Nevanlinna-Pick interpolation in several variables

## TL;DR

This paper extends commutant lifting and Nevanlinna-Pick interpolation theorems to multipliers between vector-valued Drury-Arveson space and various reproducing kernel Hilbert spaces over the unit ball, including Hardy and Bergman spaces.

## Contribution

It establishes a unified framework for commutant lifting and interpolation in several variables for a broad class of Hilbert spaces.

## Key findings

- Generalized commutant lifting theorem for vector-valued spaces
- Nevenlinna-Pick interpolation results in several variables
- Applicable to Hardy, Bergman, and weighted Bergman spaces

## Abstract

This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.