Nuclearidade de operadores integrais positivos

Abstract

This work is to presents conditions to ensure that a positive integral operator K : L2(X,a) ^ L2(X, a), given by K(f)(x) = Jx f (y)k(x,y)da(y), for all f G L2(X,ff), is nuclear, that is, to ensure that X]/gB l|K(f)|| < to, when R is an orthonormal bases to L2 (A, a). We ^^^ze the case in which X is a Lebesgue mensurable subset of Mn, by using the Mercer’s Theorem. When we can’t use this theorem, we use approximations methods by operators whose kernels satisfies such theorem, by using Hardy-Littlewood’s maximal function. In that case, the nuclearity of K is related to limits in appropriate norms. This procedure may be used to integral operators whose kernels are linear combinations of positive definite kernels. To finish this work, looking to the case where X is a cr-finite measure space, not necessarily in Mn, and endowed with a special filter, a similar method to approximate the auxiliary operators and kernels is used, without the use of Mercer’s Theorem and using martingals and maximal function instead.