  We study the moments of equilibrium measures for iterated function systems
(IFSs) and draw connections to operator theory. Our main object of study is the
infinite matrix which encodes all the moment data of a Borel measure on R^d or
C. To encode the salient features of a given IFS into precise moment data, we
establish an interdependence between IFS equilibrium measures, the encoding of
the sequence of moments of these measures into operators, and a new
correspondence between the IFS moments and this family of operators in Hilbert
space. For a given IFS, our aim is to establish a functorial correspondence in
such a way that the geometric transformations of the IFS turn into
transformations of moment matrices, or rather transformations of the operators
that are associated with them.
  We first examine the classical existence problem for moments, culminating in
a new proof of the existence of a Borel measure on R or C with a specified list
of moments. Next, we consider moment problems associated with affine and
non-affine IFSs. Our main goal is to determine conditions under which an
intertwining relation is satisfied by the moment matrix of an equilibrium
measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical
example, we study boundedness and spectral properties of moment matrices viewed
as Kato-Friedrichs operators on weighted l^2 spaces.
