  We study analytic and geometric properties of Stieltjes and inverse Stieltjes
families defined on a separable Hilbert space and establish various minimal
representations for them by means of compressed resolvents of various types of
linear relations. Also attention is paid to some new peculiar properties of
Stieltjes and inverse Stieltjes families, including an analog for the notion of
inner functions which will be characterized in an explicit manner. In addition,
families which admit different types of scale invariance properties are
described. Two transformers that naturally appear in the Stieltjes and inverse
Stieltjes classes are introduced and their fixed points are identified.

Functional Analysis