DELGADO GARRIDO , Olvido

El origen de la integración de funciones escalares respecto de una medida vectorial se remonta al año 1955, cuando Bartle, Dunford y Schwartz extienden al caso vectorial el teorema de representación de Riesz. La versión clásica de este teorema establece que todo funcional lineal y positivo T: C(K) → ℂ, donde K un espacio de Hausdorff compacto y...

In this paper we give conditions under which a positive order continuous operator T defined on a Banach function space can be extended, preserving the order continuity in a certain optimal way. The optimal domain for T turns out to be a space of weakly integrable functions with respect to a vector measure (defined on a δ-ring) canonically...

Let T be a kernel operator with values in a rearrangement invariant Banach function space X on [0,∞) and defined over simple functions on [0,∞) of bounded support. We identify the optimal domain for T (still with values in X) in terms of interpolation spaces, under appropriate conditions on the kernel and the space X. The techniques used are...

Given a vector measure ν with values in a Banach space X, we consider the space L1(ν) of real functions which are integrable with respect to ν. We prove that every order continuous Banach function space Y continuously contained in L1(ν) is generated via a certain positive map ϱ related to ν and defined on X* x M, where X* is the dual space of X...

Given a vector measure ν defined on a δ-ring with values in a Banach space, we study the relation between the analytic properties of the measure ν and the lattice properties of the space L1(ν) of real functions which are integrable with respect to ν.

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En geometría moderna, la relación entre la estructura algebraica delos elementos geométricos y la descripción física de estos mismos esa menudo demasiado abstracta para que se pueda explicar usando sólo su representación matricial. En este artículo presentamos algunas herramientas para la enseñanza de los...

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We study extension of operators T : E → L0([0, 1]), where E is an F–function space and L0([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the...

Let 1 ≤ p ≤ q < ∞ and let X be a p-convex Banach function space over a σ-finite measure μ. We combine the structure of the spaces L p(μ) and Lq (ξ ) for constructing the new space S q X p (ξ ), where ξ is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every...

In this paper, we present new results about the space Lp (ν) for ν being a vector measure defined in the Borel σ -algebra of a compact abelian group G and satisfying certain property concerning translation of simple functions. Namely, we show that Lp (ν) is a translation invariant space which can be endowed with an algebra structure via usual...

In this paper we prove that every Banach lattice having the Fatou property and having its s-order continuous part as an order dense subset, can be represented as the space L1 w(n) of weakly integrable functions with respect to some vector measure n defined on a d-ring.

Let T : X1 → Y1 and S : X2 → Y2 be two continuous linear operators between Banach function spaces related to a finite measure space. Under some lattice requirements on the spaces involved, we give characterizations by means of inequalities of when T can be strongly factorized through S, that is, T = Mg ◦ S ◦Mf with Mf : X1 → X2 and Mg : Y2 → Y1...

We study the optimal domain for the Hardy operator considered with values in a rearrangement invariant space. In particular, this domain can be represented as the space of integrable functions with respect to a vector measure defined on a δ-ring. A precise description is given for the case of the minimal Lorentz spaces.

Consider a couple of Banach function spaces X and Y over the same measure space and the space XY of multiplication operators from X into Y . In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of XY . At this end, using the "generalized K¨othe duality" for Banach function spaces, we...

Let X be a saturated Banach function space and denote by X′ its Köthe dual. In the paper (Delgado and Sánchez Pérez in Positivity 20:999–1014, 2016) referenced in the title it is implicitly used that the closed unit ball BX′ of X′ is compact for the topology σ(X′,X) on X′ defined by the elements of X. This fact could be not true in general if X...

Given a set function Λ with values in a Banach space X, we construct an integration theory for scalar functions with respect to Λ by using duality on X and Choquet scalar integrals. Our construction extends the classical Bartle–Dunford–Schwartz integration for vector measures. Since just the minimal necessary conditions on Λ are required,...

Given a Banach space valued q-concave linear operator T defined on a σ-order continuous quasi-Banach function space, we provide a description of the optimal domain of T preserving q-concavity, that is, the largest σ-order continuous quasi-Banach function space to which T can be extended as a q-concave operator. We show in this way the existence...

Let X(μ) be a function space related to a measure space (Ω,Σ, μ) with χΩ ∈ X(μ) and let T : X(μ) → E be a Banach spacevalued operator. It is known that if T is pth power factorable then the largest function space to which T can be extended preserving pth power factorability is given by the space Lp(mT) of p-integrable functions with respect to...

Consider two continuous linear operators T : X1(μ) ! Y1( ) and S : X2(μ) ! Y2( ) between Banach function spaces related to different -finite measures μ and . We characterize by means of weighted norm inequalities when T can be strongly factored through S, that is, when there exist functions g and h such that T (f) = gS(hf) for all f 2...

Let ν be a vector measure with values in a Banach space Z. The integration map Iν:L1(ν)→Z, given by f↦∫fdν for f ∈ L 1(ν), always has a formal extension to its bidual operator I∗∗ν:L1(ν)∗∗→Z∗∗. So, we may consider the "integral" of any element f ** of L 1(ν)** as I **ν(f **). Our aim is to identify when these integrals lie in more tractable...

We consider the space H(cesp)of all Dirichlet series whose coefficients belong to the Cesàro sequence space cesp, consisting of all complex sequences whose absolute Cesàro means are in p, for 1 1/q}, where 1/p +1/q=1.

Given two Banach function spaces X and Y related to a measure μ, the Y-dual space XY of X is defined as the space of the multipliers from X to Y. The space XY is a generalization of the classical Köthe dual space of X, which is obtained by taking Y = Lt(μ). Under minimal conditions, we can consider the Y-bidual space XYY of X (i.e. the Y-dual...

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...