Spectral integration from dominated ergodic estimates

Suppose that (Omega, M, mu) is a sigma-finite measure space, 1 < p < infinity, and T: L-p(mu) --> L-P(mu) is a bounded, invertible, separation-preserving linear operator Such that the two-sided ergodic means of the linear modulus of T are uniformly bounded in norm. Using the spectral structure of T, we obtain a functional calculus for T associated with the algebra of Marcinkiewicz multipliers defined on the unit circle.