Commutant lifting and interpolation on the unit ball

TL;DR

This paper advances the understanding of commutant lifting and interpolation problems within Hardy spaces and Schur functions on the unit ball, highlighting the significance of inner functions, with novel results even in the classical case.

Contribution

It provides new solutions to commutant lifting and interpolation problems on the unit ball, emphasizing the role of inner functions, including new findings in the one-variable case.

Findings

Solutions to commutant lifting and interpolation problems on the unit ball.

New insights into the role of inner functions in these problems.

Novel results related to inner functions in classical one-variable setting.

Abstract

We solve the commutant lifting and interpolation problems in the setting of the Hardy space and Schur functions on the open unit ball of $\mathbb{C}^n$. Our solutions also signify the role of inner functions on the unit ball, objects whose existence was once in doubt and to which Aleksandrov, Rudin, and others made fundamental contributions. Some of our results, particularly those related to inner functions, are new even in the classical, one-variable case.