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Abstract Interpolation Problem in Nevanlinna Classes

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Modern Analysis and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 190))

Abstract

The abstract interpolation problem (AIP) in the Schur class was posed V. Katznelson, A. Kheifets and P. Yuditskii in 1987. In the present paper an analog of the AIP for Nevanlinna classes is considered. The description of solutions of the AIP is reduced to the description of \( \mathcal{L} \)-resolvents of some model symmetric operator associated with the AIP. The latter description is obtained by using the M.G. Kreĭn’s theory of \( \mathcal{L} \)-resolvent matrices. Both regular and singular cases of the AIP are treated. The results are illustrated by the following examples: bitangential interpolation problem, full and truncated moment problems. It is shown that each of these problems can be included into the general scheme of the AIP.

This research has been done partially while the author was visiting the Department of Mathematics of Weizmann Institute of Science as a Weston Visiting Scholar.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Donetsk National University, 24 Universitetskaya str., 83055, Donetsk, Ukraine

    Vladimir Derkach

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  1. Vladimir Derkach
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Editors and Affiliations

  1. Department of Theoretical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Vadim M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    Israel Gohberg

  3. Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska st., 01601, Kyiv-4, Ukraine

    Anatoly Kochubei , Yurij Berezansky , Myroslav Gorbachuk  & Valentyna Gorbachuk , ,  & 

  4. Department of Mathematical Physics, Odessa National I.I. Mechnikov University, Dvoryanska st. 2, 65026, Odessa, Ukraine

    Gennadiy Popov

  5. Institute of Analysis and Scientific Computing, Technical University of Vienna, Wiedner Hauptstrasse 8-10, 1040, Vienna, Austria

    Heinz Langer

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Derkach, V. (2009). Abstract Interpolation Problem in Nevanlinna Classes. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 190. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9919-1_12

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