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

2011
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Transactions of the Moscow Mathematical Society
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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

doi:10.1090/s0077-1554-2012-00195-2
fatcat:fiwaumdgybg5rm3nssw4fbfzbm