No abstract

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators . . . ). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T . Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.

Abstract. We study some Banach lattice properties of the space L 1 w (ν) of weakly integrable functions with respect to a vector measure ν defined on a δ-ring. Namely, we analyze order continuity, order density and Fatou type properties. We will see that the behavior of L 1 w (ν) differs from the case in which ν is defined on a σ-algebra whenever ν does not satisfy certain local σ-finiteness property.

Abstract. We study compactness and related topological properties in the space L 1 (m) of a Banach space valued measure m when the natural topologies associated to the convergence of the vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L 1 (m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L 1 (m). The strong weakly compact generation of L 1 (m) is discussed as well.

We develop a new topological structure for the construction of a reinforcement learning model in the framework of financial markets. It is based on Lipschitz type extension of reward functions defined in metric spaces. Using some known states of a dynamical system that represents the evolution of a financial market, we use our technique to simulate new states, that we call "dreams". These new states are used to feed a learning algorithm designed to improve the investment strategy.