Operator Stieltjes integrals with respect to a spectral measure and solutions of some operator equations
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S. Albeverio and A. K. Motovilov
Translated by: V. E. Nazaikinskii - Trans. Moscow Math. Soc. 2011, 45-77
- DOI: https://doi.org/10.1090/S0077-1554-2012-00195-2
- 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.References
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Bibliographic 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
- 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. - © Copyright 2012 American Mathematical Society
- Journal: Trans. Moscow Math. Soc. 2011, 45-77
- MSC (2010): Primary 47B15; Secondary 47A56, 47A62
- DOI: https://doi.org/10.1090/S0077-1554-2012-00195-2
- MathSciNet review: 3184812