Measure preserving composition operators

We consider composition operators T induced on functional Hilbert spaces H = L₂(S, ∑, μ) byTf(·) = f(h(·)) where h: S → S is a nonsingular transformation. For these mappings T: H → H we give conditions under which they accept invariant Borel probability measures, and we relate the two structures of T, i.e., that of a bounded linear operator to that of a measure preserving transformation.