L1-spaces of vector measures defined on δ-rings

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.Ministerio de Ciencia y Tecnología BFM2003-06335-C03-0

Optimal domains for kernel operators on [0,∞)×[0,∞)

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 based on the relation between linear operators and vector measures.Ministerio de Ciencia y Tecnología BFM2003-06335-C03-0

Optimal domain for the Hardy operator

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.Ministerio de Ciencia y Tecnología (MCYT). Españ

Representation of Banach lattices as L1w spaces of a vector measure defined on a δ-ring

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.Ministerio de Educación y Ciencia MTM2009-12740-C03-02Universitat Politécnica de Valéncia PAID-10 Ref. 214

Optimal Extensions for pth Power Factorable Operators

[EN] Let be a function space related to a measure space with and let be a Banach space-valued 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 L (p) (m (T) ) of p-integrable functions with respect to m (T) , where is the vector measure associated to T via . In this paper, we extend this result by removing the restriction . In this general case, by considering m (T) defined on a certain -ring, we show that the optimal domain for T is the space . We apply the obtained results to the particular case when T is a map between sequence spaces defined by an infinite matrix.O. Delgado gratefully acknowledges the support of the Ministerio de Economia y Competitividad (Project #MTM2012-36732-C03-03) and the Junta de Andalucia (Projects FQM-262 and FQM-7276), Spain. E.A. Sanchez Perez acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Project #MTM2012-36740-C02-02), Spain.Delgado Garrido, O.; Sánchez Pérez, EA. (2016). Optimal Extensions for pth Power Factorable Operators. Mediterranean Journal of Mathematics. 13(6):4281-4303. https://doi.org/10.1007/s00009-016-0745-1S42814303136Bennett, C., Sharpley, R.: Interpolation of operators. Academic Press, Boston (1988)Brooks J.K., Dinculeanu N.: Strong additivity, absolute continuity and compactness in spaces of measures, J. Math. Anal. Appl 45, 156–175 (1974)Calabuig J.M., Delgado O., Juan M.A., Sánchez Pérez E.A.: On the Banach lattice structure of L 1 w of a vector measure on a $${\delta}$$ δ -ring. Collect. Math 65, 67–85 (2014)Calabuig J.M., Delgado J.M., Sánchez Pérez E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl 364, 88–103 (2010)Calabuig J.M., Juan M.A., Sánchez Pérez E.A.: Spaces of p-integrable functions with respect to a vector measure defined on a $${\delta}$$ δ -ring. Operators Matrices 6, 241–262 (2012)del Campo R., Fernández A., Galdames O., Mayoral F., Naranjo F.: Complex interpolation of operators and optimal domains. Integr. Equ. Oper. Theory 80, 229–238 (2014)Delgado O.: L 1-spaces of vector measures defined on $${\delta}$$ δ -rings. Arch. Math. 84, 432–443 (2005)Delgado O.: Optimal domains for kernel operators on $${[0,\infty)\times[0,\infty)}$$ [ 0 , ∞ ) × [ 0 , ∞ ) . Studia Math. 174, 131–145 (2006)Diestel, J., Uhl Jr, J.J.: Vector measures. Math. Surveys, vol. 15. American Mathematical Society, Providence (1977)Galdames Bravo O.: On the norm with respect to vector measures of the solution of an infinite system of ordinary differential equations. Mediterr. J. Math. 12, 939–956 (2015)Galdames Bravo O.: Generalized Köthe p-dual spaces. Bull. Belg. Math. Soc. Simon Stevin 21, 275–289 (2014)Galdames Bravo O., Sánchez Pérez E.A.: Optimal range theorems for operators with p-th power factorable adjoints. Banach J. Math. Anal. 6, 61–73 (2012)Galdames Bravo O., Sánchez Pérez E.A.: Factorizing kernel operators. Integr. Equ. Oper. Theory 75, 13–29 (2013)Kalton, N.J., Peck, N.T., Roberts, J.W.: An F-space Sampler. London Math. Soc. Lecture Notes, vol. 89. Cambridge University Press, Cambridge (1985)Lewis D.R: On integrability and summability in vector spaces. Ill. J. Math. 16, 294–307 (1972)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol.II. Springer, Berlin (1979)Masani P.R., Niemi H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. I. Scalar-valued measures on $${\delta}$$ δ -rings. Adv. Math. 73, 204–241 (1989)Masani P.R., Niemi H.: The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math. 75, 121–167 (1989)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal domain and integral extension of operators acting in function spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008

Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1 <= p <= q

[EN] Let and let X be a p-convex Banach function space over a -finite measure . We combine the structure of the spaces and for constructing the new space , where 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 q-summing operator T defined on X can be continuously (strongly) extended to . Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through -spaces when . Thus, our result completes the picture, showing what happens in the complementary case 1 <= p <= q.O. Delgado gratefully acknowledge the support of the Ministerio de Economia y Competitividad (project #MTM2012-36732-C03-03) and the Junta de Andalucia (projects FQM-262 and FQM-7276), Spain. E. A. Sanchez Perez acknowledges with thanks the support of the Ministerio de Economia y Competitividad (project #MTM2012-36740-C02-02), Spain.Delgado Garrido, O.; Sánchez Pérez, EA. (2016). Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1 <= p <= q. Positivity. 20(4):999-1014. https://doi.org/10.1007/s11117-016-0397-1S9991014204Calabuig, J.M., Delgado, O., Sánchez Pérez, E.A.: Factorizing operators on Banach function spaces through spaces of multiplication operators. J. Math. Anal. Appl. 364, 88–103 (2010)Calabuig, J.M., Rodríguez, J., Sánchez-Pérez, E.A.: Strongly embedded subspaces of $$p$$ p -convex Banach function spaces. Positivity 17, 775–791 (2013)Defant, A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Sánchez Pérez, E.A.: Maurey-Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Delgado, O., Sánchez Pérez, E.A.: Summability properties for multiplication operators on Banach function spaces. Integr. Equ. Oper. Theory 66, 197–214 (2010)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer-Verlag, Berlin (1979)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory: Adv. Appl., vol 180, Birkhäuser, Basel (2008)Sánchez Pérez, E.A.: Factorization theorems for multiplication operators on Banach function spaces. Integr. Equ. Oper. Theory 80, 117–135 (2014)Zaanen, A.C.: Integration, 2nd rev. ed., North-Holland, Amsterdam (1967

Summability Properties for Multiplication Operators on Banach Function 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 introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x ∈ X and y ∈ Y .Universitat Politécnica de Valencia PAID-06-08 Ref. 3093Ministerio de Educación y Ciencia MTM2006-13000-C03-0

Choquet type L-1-spaces of a vector capacity

[EN] Given a set function Lambda with values in a Banach space X, we construct an integration theory for scalar functions with respect to Lambda by using duality on Xand Choquet scalar integrals. Our construction extends the classical Bartle-Dunford-Schwartz integration for vector measures. Since just the minimal necessary conditions on Lambda are required, several L-1-spaces of integrable functions associated to Lambda appear in such a way that the integration map can be defined in them. We study the properties of these spaces and how they are related. We show that the behavior of the L-1-spaces and the integration map can be improved in the case when Xis an order continuous Banach lattice, providing new tools for (non-linear) operator theory and information sciences. (C) 2017 Elsevier B.V. All rights reserved.The first and second authors gratefully acknowledge the support of the Ministerio de Economia y Competitividad under projects MTM2015-65888-C4-1-P and MTM2016-77054-C2-1-P, respectively. The first author also acknowledges the support of the Junta de Andalucia (project FQM-7276), Spain.Delgado Garrido, O.; Sánchez Pérez, EA. (2017). Choquet type L-1-spaces of a vector capacity. Fuzzy Sets and Systems. 327:98-122. https://doi.org/10.1016/j.fss.2017.05.014S9812232