Interpolation without commutants

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 one 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 extension 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.