Multiparameter operator limit theorems

Abstract

We improve the multiparameter version of Rota’s theorem obtained by Frangos and Sucheston for bistochastic operators to the case of positive linear contractions. As a special case, we also obtain the multiparamter convergence of Burkholder–Chow type for positive self-adjoint contractions on $L_2$ spaces. The idea behind our approach can be applied to obtain a ratio multiparameter version of Doob’s theorem. We show that the weak-type maximal inequality approach to proving almost everywhere convergence makes it possible to give a positive answer to the question raised by Burkholder about positive definite self-adjoint contractions on $L_1$.