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M. Kreĭn’s Research on Semi-Bounded Operators, its Contemporary Developments, and Applications

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Book cover Modern Analysis and Applications

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

Abstract

We are going to consider the M. Kreĭn classical papers on the theory of semi-bounded operators and the theory of contractive self-adjoint extensions of Hermitian contractions, and discuss their impact and role in the solution of J. von Neumann’s problem about parametrization in terms of his formulas of all nonnegative self-adjoint extensions of nonnegative symmetric operators, in the solution of the Phillips-Kato extension problems (in restricted sense) about existence and parametrization of all proper sectorial (accretive) extensions of nonnegative operators, in bi-extension theory of non-negative operators with the exit into triplets of Hilbert spaces, in the theory of singular perturbations of nonnegative self-adjoint operators, in general realization problems (in system theory) of Stieltjes matrix-valued functions, in Nevanlinna-Pick system interpolation in the class of sectorial Stieltjes functions, in conservative systems theory with accretive main Schrödinger operator, in the theory of semi-bounded symmetric and self-adjoint operators invariant with respect to some groups of transformations. New developments and applications to the singular differential operators are discussed as well.

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Authors and Affiliations

  1. Department of Mathematical Analysis, East Ukrainian National University, 91034, Lugansk, Ukraine

    Yu. Arlinskiĭ

  2. Department of Mathematics, Niagara University, PO. BOX 2044, NY, 14109, USA

    E. Tsekanovskiĭ

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  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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To the memory of Mark Grigor’evich Kreĭn

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Arlinskiĭ, Y., Tsekanovskiĭ, E. (2009). M. Kreĭn’s Research on Semi-Bounded Operators, its Contemporary Developments, and Applications. 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_5

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