Semivariation

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Sánchez Pérez, Enrique Alfonso - One of the best experts on this subject based on the ideXlab platform.

  • Lorentz spaces of vector measures and real interpolation of operators
    'National Inquiry Services Center (NISC)', 2020
    Co-Authors: Del Campo R., Fernández A., Mayoral F., Naranjo F., Sánchez Pérez, Enrique Alfonso
    Abstract:

    [EN] Using the representation of the real interpolation of spaces of p-integrable functions with respect to a vector measure, we show new factorization theorems for p-th power factorable operators acting in interpolation couples of Banach function spaces. The recently introduced Lorentz spaces of the Semivariation of vector measures play a central role in the resulting factorization theorems. We apply our results to analyze extension of operators from classical weighted Lebesgue Lp-spaces ¿ in general with di¿erent weights ¿ that can be extended to their q-th powers. This is the case, for example, of the convolution operators defined by Lp-improving measures acting in Lebesgue Lp-spaces or Lorentz spaces. A new representation theorem for Banach lattices with a special lattice geometric property, as a space of vector measure integrable functions, is also proved.The fifth author acknowledges the support of the Ministerio de Economia y Competitividad (Spain) and FEDER under grant MTM2016-77054-C2-1-PDel Campo, R.; Fernández, A.; Mayoral, F.; Naranjo, F.; Sánchez Pérez, EA. (2020). Lorentz spaces of vector measures and real interpolation of operators. Quaestiones Mathematicae. 43(5-6):591-609. https://doi.org/10.2989/16073606.2019.1605413S591609435-

  • p-Variations of vector measures with respect to vector measures and integral representation of operators
    'Tusi Mathematical Research Group', 2015
    Co-Authors: Blasco De La Cruz, Oscar, Calabuig J. M., Sánchez Pérez, Enrique Alfonso
    Abstract:

    [EN]In this paper we provide two representation theorems for two relevant classes of operators from any p-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite p-Semivariation and p-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures.J.M. Calabuig and O. Blasco were supported by Ministerio de Economia y Competitividad (Spain) (project MTM2011-23164). E.A. Sanchez-Perez was supported by Ministerio de Economia y Competitividad (Spain) (project MTM2012-36740-C02-02).Blasco De La Cruz, O.; Calabuig, JM.; Sánchez Pérez, EA. (2015). p-Variations of vector measures with respect to vector measures and integral representation of operators. Banach Journal of Mathematical Analysis. 9(1):273-285. doi:10.15352/bjma/09-1-20S2732859

Monteiro, Giselle Antunes - One of the best experts on this subject based on the ideXlab platform.

  • On functions of bounded Semivariation
    2015
    Co-Authors: Monteiro, Giselle Antunes
    Abstract:

    The concept of bounded variation has been generalized in many ways. In the frame of functions taking values in Banach space, the concept of bounded Semivariation is a very important generalization. The aim of this paper is to provide an accessible summary on this notion, to illustrate it with an appropriate body of examples, and to outline its connection with the integration theory due to Kurzweil.Comment: 39 pages, to be submitted to Real Analysis Exchang

Nowak Marian - One of the best experts on this subject based on the ideXlab platform.

  • Operators on the space of vector-valued totally measurable functions
    Elsevier Inc., 2009
    Co-Authors: Nowak Marian
    Abstract:

    AbstractLet L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)‖Y→0 whenever a sequence of scalar functions (‖fn(⋅)‖X) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator T:B(Σ,X)→Y is σ-smooth if and only if its representing measure m:Σ→L(X,Y) is variationally semi-regular, i.e., m˜(An)→0 as An↓∅ (here m˜(A) stands for the Semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach–Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures m:Σ→L(X,Y) in terms of continuity of the corresponding operators T:B(Σ,X)→Y

Oscar Blasco - One of the best experts on this subject based on the ideXlab platform.

  • REMARKS ON THE Semivariation OF VECTOR MEASURES WITH RESPECT TO BANACH SPACES.
    2015
    Co-Authors: Oscar Blasco
    Abstract:

    Abstract. Let Lq(ν)⊗̂γqY = Lq(ν, Y) and X⊗̂∆pLp(µ) = Lp(µ,X). It is shown that any Lp(µ)-valued measure has finite L2(ν)-Semivariation with respect to the tensor norm L2(ν)⊗̂∆pLp(µ) for 1 ≤ p < ∞ and finite Lq(ν)-Semivariation with respect to the tensor norm Lq(ν)⊗̂γqLp(µ) whenever either q = 2 and 1 ≤ p ≤ 2 or q> max{p, 2}. However there exist measures with infinite Lq-Semivariation with respect to the tensor norm Lq(ν)⊗̂γqLp(µ) for any 1 ≤ q < 2. It is also shown that the measure m(A) = χA has infinite Lq-Semivariation with respect to the tensor norm Lq(ν)⊗̂γqLp(µ) if q < p. 1

Sofo Anthony - One of the best experts on this subject based on the ideXlab platform.

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