Almost everywhere convergence of multiple operator averages for affine semigroups

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 Lp 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 Lp (1 < p < ∞) 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