Transactions of the Moscow Mathematical Society

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ISSN 1547-738X(e) ISSN 0077-1554(p)

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
Posted: 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 does not have common points with the spectrum of a normal operator and that is a bounded operator, we construct a representation of a strong solution 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 . On the basis of this result, we establish sufficient conditions for the existence of a strong solution of the operator Riccati equation , where is another bounded operator.

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