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Journal of Operator Theory

Volume 84, Issue 1, Summer 2020  pp. 239-256.

Interpolation without commutants

Authors:  Oleg Szehr (1), Rachid Zarouf (2)
Author institution:(1) University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
(2) Aix-Marseille Universite, EA-4671 ADEF, ENS de Lyon, Campus Universitaire de Saint-Jerome, 40 Avenue Escadrille Normandie Niemen, 13013 Marseille, France and Department of Mathematics and Mechanics, Saint Petersburg State University, 28, Universitetski pr., St. Petersburg, 198504, Russia

Summary: 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.

Keywords: Nevanlinna-Pick interpolation, Carath\'eodory-Schur interpolation, Beurling-Sobolev spaces, Wiener algebra

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