  We offer a spectral analysis for a class of transfer operators. These
transfer operators arise for a wide range of stochastic processes, ranging from
random walks on infinite graphs to the processes that govern signals and
recursive wavelet algorithms; even spectral theory for fractal measures. In
each case, there is an associated class of harmonic functions which we study.
And in addition, we study three questions in depth:
  In specific applications, and for a specific stochastic process, how do we
realize the transfer operator $T$ as an operator in a suitable Hilbert space?
And how to spectral analyze $T$ once the right Hilbert space $\mathcal{H}$ has
been selected? Finally we characterize the stochastic processes that are
governed by a single transfer operator.
  In our applications, the particular stochastic process will live on an
infinite path-space which is realized in turn on a state space $S$. In the case
of random walk on graphs $G$, $S$ will be the set of vertices of $G$. The
Hilbert space $\mathcal{H}$ on which the transfer operator $T$ acts will then
be an $L^{2}$ space on $S$, or a Hilbert space defined from an energy-quadratic
form.
  This circle of problems is both interesting and non-trivial as it turns out
that $T$ may often be an unbounded linear operator in $\mathcal{H}$; but even
if it is bounded, it is a non-normal operator, so its spectral theory is not
amenable to an analysis with the use of von Neumann's spectral theorem. While
we offer a number of applications, we believe that our spectral analysis will
have intrinsic interest for the theory of operators in Hilbert space.
