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

S. Albeverio, A. K. Motovilov
2011 Transactions of the Moscow Mathematical Society  
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
more » ... he 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.
doi:10.1090/s0077-1554-2012-00195-2 fatcat:fiwaumdgybg5rm3nssw4fbfzbm