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Graph Subspaces and the Spectral Shift Function

Published online by Cambridge University Press:  20 November 2018

Sergio Albeverio ,
Konstantin A. Makarov  and
Alexander K. Motovilov
Sergio Albeverio
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany, website:, http://wiener.iam.uni-bonn.de/albeverio/albeverio.html e-mail: albeverio@uni-bonn.de
Konstantin A. Makarov
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA, website:, http://www.math.missouri.edu/people/kmakarov.html e-mail: makarov@math.missouri.edu
Alexander K. Motovilov
Affiliation:
Bogoliubov Laboratory of Theoretical Physics, JINR, Joliot-Curie str. 6, 141980 Dubna, Russia, website:, http://thsun1.jinr.ru/˜motovilv e-mail: motovilv@thsun1.jinr.ru
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Abstract

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We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.

Keywords

47B4447A1047A2047A40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 3 , 01 June 2003 , pp. 449 - 503
Copyright
Copyright © Canadian Mathematical Society 2003

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