Factorization Theorems for Multiplication Operators on Banach Function Spaces

[EN] Let X Y and Z be Banach function spaces over a measure space . Consider the spaces of multiplication operators from X into the Kothe dual Y' of Y, and the spaces X (Z) and defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey-Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.The author was supported by the Ministerio de Econom´ıa y Competitividad (Spain) under grant #MTM2012-36740-C02-02.Sánchez Pérez, EA. (2014). Factorization Theorems for Multiplication Operators on Banach Function Spaces. Integral Equations and Operator Theory. 80(1):117-135. https://doi.org/10.1007/s00020-014-2169-S11713580

Factorization through Lorentz spaces for operators acting in Banach function spaces

[EN] We show a factorization through Lorentz spaces for Banach-space-valued operators defined in Banach function spaces. Although our results are inspired in the classical factorization theorem for operators from Ls-spaces through Lorentz spaces Lq,1 due to Pisier, our arguments are different and essentially connected with Maurey's theorem for operators that factor through Lp-spaces. As a consequence, we obtain a new characterization of Lorentz Lq,1-spaces in terms of lattice geometric properties, in the line of the (isomorphic) description of Lp-spaces as the unique ones that are p-convex and p-concave.Funding was provided by Secretaria de Estado de Investigacion, Desarrollo e Innovacion and FEDER (Grant No. MTM2016-77054-c2-1-P).Sánchez Pérez, EA. (2019). Factorization through Lorentz spaces for operators acting in Banach function spaces. Positivity. 23(1):75-88. https://doi.org/10.1007/s11117-018-0593-2S7588231Achour, D., Mezrag, L.: Factorisation des opèrateurs sous-linéaires par $$L^{p,\infty }(\varOmega , \nu )$$ L p , ∞ ( Ω , ν ) et $$L^{q,1} (\varOmega ,\nu )$$ L q , 1 ( Ω , ν ) . Ann. Sci. Math. Québec. 29, 109–121 (2002)Berg, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Heidelberg (1976)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi-Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Philos. Soc. 146, 57–66 (2009)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. 61, 183–200 (1993)Krivine, J.L.: Théorèmes de factorisation dans les espaces réticulés. Séminaire d’analyse fonctionelle Maurey-Schwartz 1973–1974. Exposés XXII et XXIII. p.1–22. École Polytechnique, Paris (1974)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Mastyło, M., Sánchez Pérez, E.A.: Factorization of operators through Orlicz spaces. Bull. Malays. Math. Sci. Soc. 40, 1653–1675 (2017)Mastyło, M., Szwedek, R.: Interpolative construction and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Maurey, B.: Theorémes de factorisation pour les opèrateurs linéaires à valeurs dans les spaces Lp. Séminaire d’analyse fonctionelle Maurey-Schwartz. 1972–1973. Exposés XVII, p.1–5. École Polytechnique, Paris (1973)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Birkhäuser, Basel (2008)Pisier, G.: Factorization of operators through $$L_{p\infty }$$ L p ∞ or $$L_{p1}$$ L p 1 and noncommutative generalizations. Math. Ann. 276, 105–136 (1986)Rosenthal, H.P.: On subspaces of $$L^{p}$$ L p . Ann. Math. 97, 344–373 (1973

Social Security and the search behaviour of workers approaching retirement

This paper explores the links between unemployment, retirement and their associated public insurance programs. It is a contribution to a growing body of literature focused on a better understanding of the labor behavior of advanced-age workers, which has gained importance as the pension crisis looms. It also contributes to the literature of optimal unemployment insurance by exploring the interaction of unemployment benefits and retirement pensions. The analysis combines the development of a new theoretical model and a detailed exploration of the empirical regularities using the Spanish Muestra Continua de Vidas Laborales (MCVL) dataset. The model is an extension of the standard search model, designed to reproduce the non-stationary environment faced by workers of advanced ages (in the age range 50/65). Via calibrated simulations we show that the basic empirical re-employment and retirement patterns can be considered as rational responses to both the labor market conditions and the institutional incentives. Generous Unemployment Benefits (for durations of up to two years) together with very significant early retirement penalties, make optimal to stay unemployed without searching for large groups of unemployed workers. This moral hazard problem can be substantially alleviated through institutional reform. We explore several potential reforms and find that changing the details of early retirement pensions seems more promising than changing the Unemployment Benefit system.Unemployment, Retirement, Search models

Un nombre nuevo en la pintura del siglo XVI español.

Sin resume

Prions: an evolutionary perspective

Studies in both prion-due diseases in mammals and some non-Mendelian hereditary processes in yeasts have demonstrated that certain proteins are able to transmit structural information and self-replication. This induces the corresponding conformational changes in other proteins with identical or similar sequences. This ability of proteins may have been very useful during prebiotic chemical evolution, prior to the establishment of the genetic code. During this stage, proteins (proteinoids) must have molded and selected their structural folding units through direct interaction with the environment. The proteinoids that acquired the ability to propagate their conformations (which we refer to as conformons) would have acted as reservoirs and transmitters of a given structural information and hence could have acted as selectors for conformational changes. Despite the great advantage that arose from the establishment of the genetic code, the ability to propagate conformational changes did not necessarily disappear. Depending on the degree of involvement of this capacity in biological evolution, we propose two not mutually exclusive hypotheses: (i) extant prions could be an atavism of ancestral conformons, which would have co-evolved with cells, and (ii) the evolution of conformons would have produced cellular proteins, able to transmit structural information, and, in some cases, participating in certain processes of regulation and epigenesis. Therefore, prions could also be seen as conformons of a conventional infectious agent (or one that co-evolved with it independently) that, after a longer or shorter adaptive period, would have interacted with conformons from the host cells

Local compactness in right bounded asymmetric normed spaces

[EN] We characterize the ¿nite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, such as compactness and strong compactness. In contrast with some results found in the ex-isting literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are ¿nite dimen-sional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open ques-tion on the topology of ¿nite dimensional asymmetric normed spaces. In the positive direction, we will prove that a ¿nite dimensional asym-metric normed space is strongly locally compact if and only if it is right bounded.The first author has been supported by Conacyt grant 252849 (Mexico) and by PAPIIT grant IA104816 (UNAM, Mexico). The second author has been supported by Ministerio de Economia y Competitividad (Spain) (project MTM2016-77054-C2-1-P)Jonard Pérez, N.; Sánchez Pérez, EA. (2018). Local compactness in right bounded asymmetric normed spaces. Quaestiones Mathematicae. 41(4):549-563. https://doi.org/10.2989/16073606.2017.1391351S54956341

Positively norming sets in Banach function spaces

The notion of positively norming set, a specific definition of norming type sets for Banach lattices, is analyzed. We show that the size of positively norming sets (in terms of compactness and order boundedness) is directly related to the existence of lattice copies of L-1-spaces. As an application, we provide a version of Kadec-Pelczynski's dichotomy for order continuous Banach function spaces. A general description of positively norming sets using vector measure integration is also given.This research was supported by the Ministerio de Economia y Competitividad under project MTM2012-36740-c02-02 (Spain) (to E. A. S. P.) and by the Ministerio de Economia y Competitividad under projects MTM2010-14946, MTM2012-31286 and Grupo UCM 910346 (to P.T.)Sánchez Pérez, EA.; Tradacete Pérez, P. (2014). Positively norming sets in Banach function spaces. Quarterly Journal of Mathematics. 65(3):1049-1068. doi:10.1093/qmath/hat035S1049106865

(p,q)-Regular operators between Banach lattices

[EN] We study the class of (p,q)-regular operators between quasi-Banach lattices. In particular, a representation of this class as the dual of a certain tensor norm for Banach lattices is given. We also provide some factorization results for (p,q)-regular operators yielding new Marcinkiewicz-Zygmund type inequalities for Banach function spaces. An extension theorem for (q,)-regular operators defined on a subspace of Lq is also given.E. A. Sanchez Perez gratefully acknowledges support of Spanish Ministerio de Economia, Industria y Competitividad and FEDER under Project MTM2016-77054-C2-1-P. P. Tradacete gratefully acknowledges support of Spanish Ministerio de Economia, Industria y Competitividad through Grants MTM2016-76808-P and MTM2016-75196-P, the "Severo Ochoa Programme for Centres of Excellence in R&D" (SEV-2015-0554), and Grupo UCM 910346. The authors wish to thank the anonymous referee for his/her careful reading of the manuscript.Sánchez Pérez, EA.; Tradacete Pérez, P. (2019). (p,q)-Regular operators between Banach lattices. Monatshefte für Mathematik. 188(2):321-350. https://doi.org/10.1007/s00605-018-1247-yS3213501882Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006) (Reprint of the 1985 original)Bukhvalov, A.V.: On complex interpolation method in spaces of vector-functions and generalized Besov spaces. Dokl. Akad. Nauk SSSR 260(2), 265–269 (1981)Bukhvalov, A.V.: Order-bounded operators in vector lattices and spaces of measurable functions. Translated in J. Soviet Math. 54(5), 1131–1176 (1991). Itogi Nauki i Tekhniki, Mathematical analysis, Vol. 26 (Russian), 3–63, 148, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1988)Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Danet, N.: Lattice $$(p, q)$$ ( p , q ) -summing operators and their conjugates. Stud. Cerc. Mat. 40(1), 99–107 (1988)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland, Amsterdam (1993)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297(2), 771–790 (2004)Defant, A., Junge, M.: Best constants and asymptotics of Marcinkiewicz–Zygmund inequalities. Stud. Math. 125(3), 271–287 (1997)Gasch, J., Maligranda, L.: On vector-valued inequalities of the Marcinkiewicz–Zygmund. Herz Kriv. Type Math. Nachr. 167, 95–129 (1994)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)Kalton, N.J.: Convexity conditions for non-locally convex lattices. Glasg. Math. J 25, 141–152 (1984)Krivine, J.L.: Thèorèmes de factorisation dans les espaces rèticulès. Sèminaire Maurey–Schwartz 1973–1974: Espaces $$L^{p}$$ L p , applications radonifiantes et gèomètrie des espaces de Banach, Exp. Nos. 22 et 23. Centre de Math., Ècole Polytech., Paris (1974)Kusraev, A.G.: Dominated Operators. Mathematics and Its Applications, vol. 519. Kluwer, Dordrecht (2000)Levy, M.: Prolongement d’un opérateur d’un sous-espace de $$L_1(\mu )$$ L 1 ( μ ) dans $$L_1(\nu )$$ L 1 ( ν ) . Seminar on Functional Analysis, 1979–1980, Exp. No. 5, 5 pp., École Polytech., Palaiseau (1980)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II: Function Spaces. Springer, Berlin (1979)Nielsen, N.J., Szulga, J.: $$p$$ p -Lattice summing operators. Math. Nachr. 119, 219–230 (1984)Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980)Pisier, G.: Complex interpolation and regular operators between Banach lattices. Arch. Math. (Basel) 62(3), 261–269 (1994)Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. 49(2), 237–323 (2012)Popa, N.: Uniqueness of the symmetric structure in $$L_p(\mu )$$ L p ( μ ) for $$0 < p < 1$$ 0 < p < 1 . Rev. Roum. Math. Pures Appl. 27, 1061–1083 (1982)Raynaud, Y., Tradacete, P.: Calderón–Lozanovskii interpolation of quasi-Banach lattices. Banach J. Math. Anal. 12(2), 294–313 (2018)Schep, A.R.: Products and factors of Banach function spaces. Positivity 14, 301–319 (2010)Wojtaszczyk, P.: Banach Spaces for Analysts, vol. 25. Cambridge University Press, Cambridge (1996

p-Regularity and Weights for Operators Between L-p-Spaces

[EN] We explore the connection between p-regular operators on Banach function spaces and weighted p-estimates. In particular, our results focus on the following problem. Given finite measure spaces mu and nu, let T be an operator defined from a Banach function space X(nu) and taking values on L-p(vd mu) for v in certain family of weights V subset of L-1(mu)+ we analyze the existence of a bounded family of weights W subset of L-1(nu)+ such that for every v is an element of V there is w is an element of W in such a way that T : L-p(wd nu) -> L-p (vd mu) is continuous uniformly on V. A condition for the existence of such a family is given in terms of p-regularity of the integration map associated to a certain vector measure induced by the operator T.E. A. Sanchez Perez gratefully acknowledges support of Spanish Ministerio de Ciencia, Innovacion y Universidades, Agencia Estatal de Investigacion and FEDER through grant MTM2016-77054-C2-1-P. P. Tradacete gratefully acknowledges support of Spanish Ministerio de Economa, Industria y Competitividad, Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) through grants MTM2016-76808-P (AEI/FEDER, UE), MTM2016-75196-P (AEI/FEDER, UE) and the \Severo Ochoa Programme for Centres of Excellence in R&D"(SEV-2015-0554).Sánchez Pérez, EA.; Tradacete, P. (2020). p-Regularity and Weights for Operators Between L-p-Spaces. Zeitschrift für Analysis und ihre Anwendungen. 39(1):41-65. https://doi.org/10.4171/ZAA/1650S416539