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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

Operator Stieltjes integrals with respect to a spectral measure and solutions of some operator equations


Authors: S. Albeverio and A. K. Motovilov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 72 (2011), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2011, 45-77
MSC (2010): Primary 47B15; Secondary 47A56, 47A62
Published electronically: January 12, 2012
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Abstract: We introduce the notion of Stieltjes integral with respect to the spectral measure corresponding to a normal operator. Sufficient conditions for the existence of this integral are given, and estimates for its norm are established. The results are applied to operator Sylvester and Riccati equations. Assuming that the spectrum of a closed densely defined operator $ A$ does not have common points with the spectrum of a normal operator $ C$ and that $ D$ is a bounded operator, we construct a representation of a strong solution $ X$ of the Sylvester equation $ X\mspace {-2mu}A-CX=D$ in the form of an operator Stieltjes integral with respect to the spectral measure of $ C$. On the basis of this result, we establish sufficient conditions for the existence of a strong solution of the operator Riccati equation $ YA-CY+YBY=D$, where $ B$ is another bounded operator.


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Additional Information

S. Albeverio
Affiliation: Institut für angewandte Mathematik, Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany
Email: albeverio@uni-bonn.de

A. K. Motovilov
Affiliation: Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow Region, Russian Federation
Email: motovilv@theor.jinr.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-2012-00195-2
PII: S 0077-1554(2012)00195-2
Keywords: Operator Stieltjes integral, operator-valued function, normal operator, spectral measure, Sylvester equation, Riccati equation
Published electronically: January 12, 2012
Additional Notes: Supported by the Russian Foundation for Basic Research, Deutsche Forschungsgemeinschaft, and the Heisenberg–Landau Program.
A. K. Motovilov is grateful to the Institute for Applied Mathematics, University of Bonn, for kind hospitality when carrying out this research.
Article copyright: © Copyright 2012 American Mathematical Society