Spectral integration from dominated ergodic estimates

Abstract

Suppose that $(\Omega,\mathcal{M},\mu)$ is a $\sigma$-finite measure space, $1 \lt p \lt \infty$, and $T: L^{p}(\mu) \rightarrow L^{p}(\mu)$ 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$\ldots$