Commutant lifting, interpolation, and perturbations on the polydisc

TL;DR

This paper extends commutant lifting theory to the polydisc, providing classifications, solving the Nevanlinna-Pick interpolation problem, and addressing perturbation issues for bounded analytic functions.

Contribution

It introduces two classifications of commutant lifting in several variables and solves key interpolation and perturbation problems on the polydisc.

Findings

Classified commutant lifting via linear functional contractivity.

Developed a distance formula for commutant lifting.

Solved the Nevanlinna-Pick interpolation problem on the polydisc.

Abstract

The fundamental theorem on commutant lifting due to Sarason does not carry over to the setting of the polydisc. This paper presents two classifications of commutant lifting in several variables. The first classification links the lifting problem to the contractivity of certain linear functionals. The second one transforms it into nonnegative real numbers via a distance formula. We also solve the Nevanlinna-Pick interpolation problem for bounded analytic functions on the polydisc. Along the way, we solve a perturbation problem for bounded analytic functions. Commutant lifting and interpolation on the polydisc solve two well-known problems in Hilbert function space theory.