Interpolation without commutants

TL;DR

This paper presents a novel dual-space approach for interpolation in Banach spaces of holomorphic functions, providing a new method to compute minimal interpolant norms without relying on Hilbert space structures.

Contribution

It introduces a dual-space method that extends functionals without increasing their norm, offering an alternative to the commutant lifting approach in interpolation theory.

Findings

Derived explicit formulas for minimal interpolant norms.

Extended interpolation results to Beurling-Sobolev spaces.

Provided a new proof technique using functional extensions.

Abstract

We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carath\'eodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting approach of D. Sarason and B. Nagy-C.Foia\c{s}. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This Functional extensions lemma plays a similar role as Sarason's Commutant lifting theorem but it only involves the predual of X and no Hilbert space structure is needed. As an example, we present the respective Pick-type interpolation theorems for Beurling-Sobolev spaces.