A Determination of the Existence of Various Types of Positive Systems in L^p

Abstract

We consider various types of generalized bases in spaces of the type L^p(T), where T=[0,1]. More specifically, we determine whether there exists a system {f_n}_n, of the type under consideration, with the property f_n(t)>=0 almost everywhere, for each n in the natural numbers. We refer to a system with the property of almost everywhere non-negativity, as a positive system.

In the spaces with 1<= p < infinity, we determine that there do not exist positive unconditional Schauder bases, and positive unconditional quasibases. In the aforementioned spaces, we determine that there do exist positive conditional quasibases, positive conditional pseudobases, and positive exact systems. In the spaces with 1< p < infinity, we determine that there do not exist positive monotone bases, and that there do exist positive exact systems with exact dual systems. In L^2(T), we demonstrate that there do not exist positive frames, positive orthonormal bases, and positive Riesz bases. Finally, in the spaces with 0< p <= infinity, we show that there do exist positive Hamel bases. Secondary considerations explore product systems on the spaces L^p(T^2).