  This paper presents Stieltjes-type integration for operator-valued functions
with respect to spectral families. The relation between Riemann-Stieltjes
integrals associated with some classes of spectral families including, in
particular, those that arise in the context of the Stone representation theorem
for Banach spaces is established. The developed approach is applied to the
structural analysis of the operators commuting with a one-parameter
representation satisfying the Generalized Stone theorem.
