Operator limit theorems
1966; American Mathematical Society; Volume: 121; Issue: 1 Linguagem: Inglês
10.1090/s0002-9947-1966-0190757-4
ISSN1088-6850
Autores Tópico(s)Approximation Theory and Sequence Spaces
ResumoNORTON STARR(i) 0. Introduction.In this paper certain almost everywhere convergence and maximal theorems are proved, extending a type of limit theorem developed by Rota [22] and generalized by Doob [9].Given a sequence {T"}™=1 of linear operators on Lp(X,lZ,p) into itself for some pe [1, oo) and some totally ¡j-finite positive measure space (X,~L,p), we say the T" satisfy a maximal theorem if jx(sup\Tf\J dp^Apjjf\"dp, for some constant Ap and all/e Lp(X,T,,p).We say a sequence {/"}"= x of functions in Lp(X,l,,p) converges boundedly in LP if lim"_ xfn exists almost everywhere and j(sup"|/"|)p^<°o.Among the actively developed theories of bounded convergence are the ergodic theory, martingale theory, singular integral theory, the behavior of monotone sequences of operators [13], and the study of powers of self-adjoint positive definite operators [5], [24], [1].The starting point for our investigations is the following new type of limit theorem due to Rota [22]: given doubly stochastic operators S", n = 1,2,3, ••■, on functions integrable over a probability space, limk^^SfS*■ ■ ■ SkSkSk-x---Sxf exists a.e. and boundedly in Lpfor any/eLp(l < p < oo).In deriving our results, we apply the techniques used by Doob [9] in proving a generalization, in the form of a ratio theorem, of the above theorem.We remark that for the special case in which all the operators Sk, S*ot Rota's theorem coincide with a single operator Tof the form T= Ex • E2, where each E; is a conditional expectation, bounded almost everywhere convergence in L2 was obtained earlier by Burkholder and Chow [5].As a preliminary to this result, Burkholder and Chow actually obtained bounded almost everywhere convergence in L2 of T2nf, assuming only that T be a self-adjoint operator on L2 which maps
Referência(s)