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

摘要

Suppose that (Ω,μ) is a σ-finite measure space, and 1 < p < ∞. Let T:L~p(μ → L~p(μ) be a bounded invertible linear operator such that T and T~(-1) are positive. Denote by E_n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {T~j+T~(-j)}_(j=1)~∞. Martín-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 E_n(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 {T~j+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 E_n(T)_(n=1)~∞.