Mathematics – Spectral Theory
Scientific paper
2004-10-27
Trudy Moskovskogo Matematicheskogo Obshchestva 72:1 (2011), 63-103 (Russian); English translation: Transactions of the Moscow
Mathematics
Spectral Theory
Scientific paper
10.1090/S0077-1554-2012-00195-2
We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the entry A is separated from the spectrum of C, and D is a bounded operator, we obtain a representation for the strong solution X to the Sylvester equation XA-CX=D in the form of an operator Stieltjes integral with respect to the spectral measure of C. By using this result we then establish sufficient conditions for the existence of a strong solution to the operator Riccati equation YA-CY+YBY=D where B is another bounded operator.
Albeverio Sergio
Motovilov Alexander K.
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