ACTA issues

Almost everywhere convergence of multiple operator averages for affine semigroups

Takeshi Yoshimoto

Acta Sci. Math. (Szeged) 84:3-4(2018), 509-554

Abstract. This paper projects another affine case study in the program of analyzing multiparameter a.e. convergence, based on the Sucheston's type convergence principles. An affine semigroup is considered as a natural extension of strongly continuous semigroups of linear operators on $L_{p}$ spaces. We prove some affine extensions of multiparameter martingale theorems, multiparameter ergodic theorems, and multiparameter ergodic theorems for the so-called nonlinear sums. Moreover, an affine (nonlinear) generalization is given of Berkson--Bourgain--Gillespie's theorem concerning the connection between the ergodic Hilbert transform and the ergodic theorem for power-bounded invertible linear operators on $L_{p}$ ($1< p< \infty $) spaces. In addition, the random ergodic Hilbert transforms will be established. We improve the local ergodic theorem of McGrath concerning strongly continuous $m$-parameter semigroups of positive linear operators in a more general affine setting. We shall also show that the Sucheston convergence principle is also very effective even in yielding a multiparameter generalization of Starr's theorem.The final section includes some examples.

DOI: 10.14232/actasm-016-510-0

AMS Subject Classification (1991): 47A35, 40H05; 40G10

Keyword(s): affine semigroup, compound semigroup, ergodic Hilbert transform, random ergodic Hilbert transform, Cotlar's theorem, Berkson-Bourgain-Gillespie's theorem, Sucheston's type convergence principle, Orlicz class, multiparameter martingale theorem, nonlinear sum, ergodic theorem for affine semigroups, Abelian ergodic theorem for affine semigroups, Starr's theorem

Received February 15, 2016 and in final form February 26, 2018. (Registered under 10/2016.)