Interpolation without commutants

Oleg Szehr , Rachid Zarouf (1)

1 ADEF - Apprentissage, Didactique, Evaluation, Formation

We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carathéodory-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.Foiaş. 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.

Domaines: Analyse fonctionnelle [math.FA]; Théorie spectrale [math.SP]; Variables complexes [math.CV]

Oleg Szehr, Rachid Zarouf. Interpolation without commutants. Journal of Operator Theory, In press, 84 (1), pp.18. ⟨10.7900/jot.2019may21.2264⟩. ⟨hal-02898954⟩