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

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 $XA-CX=D$ 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.