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Interpolation without commutants

Abstract : 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.
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Submitted on : Tuesday, July 14, 2020 - 12:32:30 PM
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Oleg Szehr, Rachid Zarouf. Interpolation without commutants. Journal of Operator Theory, Theta Foundation, In press, 84 (1), pp.18. ⟨10.7900/jot.2019may21.2264⟩. ⟨hal-02898954⟩



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