A Tauberian Theorem for Ergodic Averages, Spectral Decomposability, and the Dominated Ergodic Estimate for Positive Invertible Operators

Abstract

Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:Lp(μ → L p(μ) be a bounded invertible linear operator such that T and T−1 are positive. Denote by \({\mathfrak{E}}\)n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {Tj+T−j}j =1∞. Martin-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of En(T)n=1∞; (b) a uniform App estimate in terms of discrete weights generated by the pointwise action on Ω of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {Tj+T−j}j=1∞ already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform Ap weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages En(T)n = 1∞.