  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.
