Interpolation without commutants
Functional Analysis
2020-07-16 v1 Complex Variables
Spectral Theory
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.
Keywords
Cite
@article{arxiv.2007.07597,
title = {Interpolation without commutants},
author = {Oleg Szehr and Rachid Zarouf},
journal= {arXiv preprint arXiv:2007.07597},
year = {2020}
}