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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.

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