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Spectral integration from dominated ergodic estimates. (English) Zbl 0930.42004
Suppose that $$(\Omega,{\mathcal M},\mu)$$ is a $$\sigma$$-finite measure space, $$1<p<\infty$$, and $$T: L^p(\mu)\to L^p(\mu)$$ 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.

##### MSC:
 42A45 Multipliers in one variable harmonic analysis 47B40 Spectral operators, decomposable operators, well-bounded operators, etc. 42B15 Multipliers for harmonic analysis in several variables 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47A35 Ergodic theory of linear operators 28D05 Measure-preserving transformations