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A Functional Model for the Lie Algebra SL(2, ℝ) of Linear Non-self-adjoint Operators

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Book cover Operator Theory, System Theory and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 123))

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Abstract

An important task of the spectral analysis is to construct the model representations for linear operators realizing them via the operator of multiplication by an independent variable in a particular function space. Unlike those spectral decompositions for self-adjoint (unitary) operators constructed by von Neumann, similar representations for non-self-adjoint (non-unitary) operators are rather arduous to obtain. In the fifties Livšic [20] undertook investigations in this direction and devised a theory of characteristic functions and a theory of triangular models of linear operators. Later, in mid-sixties, a theory of dilations for semigroups of contractions was created by Sz.-Nagy and Foias [1]. Simultaneously, Lax and Phillips [2] had shaped a geometric theory of scattering of acoustic waves at bounded obstacles. Further development in these three areas supplied a basis for creating a method of studying non-self-adjoint operators and building correspondent models.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Kharkov State University, Kharkov, Ukraine, 310077

    V. A. Zolotarev

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  1. V. A. Zolotarev
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Editors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 84105, Israel

    Daniel Alpay  & Victor Vinnikov  & 

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Zolotarev, V.A. (2001). A Functional Model for the Lie Algebra SL(2, ℝ) of Linear Non-self-adjoint Operators. In: Alpay, D., Vinnikov, V. (eds) Operator Theory, System Theory and Related Topics. Operator Theory: Advances and Applications, vol 123. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8247-7_25

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  • DOI: https://doi.org/10.1007/978-3-0348-8247-7_25

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9491-3

  • Online ISBN: 978-3-0348-8247-7

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