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

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

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

(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

Ideals of multilinear mappings via Orlicz spaces and translation invariant operators

This is the peer reviewed version of the following article: Mastylo, M, Sánchez Pérez, EA. Ideals of multilinear mappings via Orlicz spaces and translation invariant operators. Mathematische Nachrichten. 2021; 294: 959-976, which has been published in final form at https://doi.org/10.1002/mana.201900380. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] We study some new summability properties of multilinear operators. We introduce the concepts of phi-summing, phi semi-integral and phi-dominated multilinear maps generated by Orlicz functions. We prove a variant of Pietsch's domination theorem for phi-summing operators, providing also a characterization of phi-dominated operators in terms of factorizations. We analyze vector-valued inequalities associated to these maps, which are applied to obtain general variants of multiple summing operators. We also study translation invariant multilinear operators acting in products of spaces of continuous functions, proving that a factorization theorem can be obtained for them as a consequence of a suitable representation of the corresponding normalized Haar measure.Ministerio de Ciencia, Innovacion y Universidades, AgenciaEstatal de Investigacion (Spain) and FEDER, Grant/Award Number: MTM2016-77054-C2-1P; The National ScienceCentre of Poland, Grant/Award Number: 2015/17/B/ST1/00064Mastylo, M.; Sánchez Pérez, EA. (2021). Ideals of multilinear mappings via Orlicz spaces and translation invariant operators. Mathematische Nachrichten. 294(5):956-976. https://doi.org/10.1002/mana.201900380956976294

Eigenmeasures and stochastic diagonalization of bilinear maps

[EN] A new stochastic approach is presented to understand general spectral type problems for (not necessarily linear) functions between topological spaces. In order to show its potential applications, we construct the theory for the case of bilinear forms acting in couples of a Banach space and its dual. Our method consists of using integral representations of bilinear maps that satisfy particular domination properties, which is shown to be equivalent to having a certain spectral structure. Thus, we develop a measure-based technique for the characterization of bilinear operators having a spectral representation, introducing the notion of eigenmeasure, which will become the central tool of our formalism. Specific applications are provided for operators between finite and infinite dimensional linear spaces.Ministerio de Ciencia, Innovacion y Universidades; Agencia Estatal de investigacion; FEDER, Grant/Award Number: MTM2016-77054-C2-1-PErdogan, E.; Sánchez Pérez, EA. (2021). Eigenmeasures and stochastic diagonalization of bilinear maps. Mathematical Methods in the Applied Sciences. 44(6):5021-5039. https://doi.org/10.1002/mma.70855021503944

Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

[EN] A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.This research was partially supported by the Grant PID2020-112759GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe".Erdogan, E.; Sánchez Pérez, EA. (2022). Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces. Mathematics. 10(2):1-24. https://doi.org/10.3390/math1002022012410

Topological Dual Systems for Spaces of Vector Measure p-Integrable Functions

[EN] We show a Dvoretzky-Rogers type theorem for the adapted version of the q-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.This work was supported by the Ministerio de Economia y Competitividad (Spain) under Grants MTM2015-66823-C2-2-P (P. Rueda) and MTM2012-36740-C02-02 (E. A. Sanchez Perez).Rueda, P.; Sánchez Pérez, EA. (2016). Topological Dual Systems for Spaces of Vector Measure p-Integrable Functions. Journal of Function Spaces. 1-8. https://doi.org/10.1155/2016/3763649S18Pérez, E. A. S. (2004). Vector measure duality and tensor product representations of $L_p$-spaces of vector measures. Proceedings of the American Mathematical Society, 132(11), 3319-3326. doi:10.1090/s0002-9939-04-07521-5Lewis, D. (1970). Integration with respect to vector measures. Pacific Journal of Mathematics, 33(1), 157-165. doi:10.2140/pjm.1970.33.157Lewis, D. R. (1972). On integrability and summability in vector spaces. Illinois Journal of Mathematics, 16(2), 294-307. doi:10.1215/ijm/1256052286Curbera, G. P. (1995). Banach Space Properties of L 1 of a Vector Measure. Proceedings of the American Mathematical Society, 123(12), 3797. doi:10.2307/2161909Ferrando, I. (2011). Factorization theorem for 1-summing operators. Czechoslovak Mathematical Journal, 61(3), 785-793. doi:10.1007/s10587-011-0027-9Fernández, A., Mayoral, F., Naranjo, F., Sáez, C., & Sánchez-Pérez, E. A. (2006). Spaces of p-integrable Functions with Respect to a Vector Measure. Positivity, 10(1), 1-16. doi:10.1007/s11117-005-0016-zOkada, S., & Ricker, W. J. (1995). The range of the integration map of a vector measure. Archiv der Mathematik, 64(6), 512-522. doi:10.1007/bf01195133Okada, S., Ricker, W. J., & Rodríguez-Piazza, L. (2002). Compactness of the integration operator associated with a vector measure. Studia Mathematica, 150(2), 133-149. doi:10.4064/sm150-2-3Okada, S., Ricker, W. J., & Rodríguez-Piazza, L. (2011). Operator ideal properties of vector measures with finite variation. Studia Mathematica, 205(3), 215-249. doi:10.4064/sm205-3-2FERRANDO, I., & SÁNCHEZ PÉREZ, E. A. (2009). TENSOR PRODUCT REPRESENTATION OF THE (PRE)DUAL OF THE Lp-SPACE OF A VECTOR MEASURE. Journal of the Australian Mathematical Society, 87(2), 211-225. doi:10.1017/s1446788709000196Ferrando, I., & Rodríguez, J. (2008). The weak topology on Lp of a vector measure. Topology and its Applications, 155(13), 1439-1444. doi:10.1016/j.topol.2007.12.014Galaz-Fontes, F. (2010). The dual space of L p of a vector measure. Positivity, 14(4), 715-729. doi:10.1007/s11117-010-0071-yRueda, P., & Sánchez-Pérez, E. A. (2015). Compactness in spaces of p-integrable functions with respect to a vector measure. Topological Methods in Nonlinear Analysis, 45(2), 641. doi:10.12775/tmna.2015.030Rueda, P., & Sánchez-Pérez, E. A. (2014). Factorization Theorems for Homogeneous Maps on Banach Function Spaces and Approximation of Compact Operators. Mediterranean Journal of Mathematics, 12(1), 89-115. doi:10.1007/s00009-014-0384-3S�nchez P�rez, E. A. (2003). Vector measure orthonormal functions and best approximation for the 4-norm. Archiv der Mathematik, 80(2), 177-190. doi:10.1007/s00013-003-0450-8Okada, S., Ricker, W. J., & Pérez, E. A. S. (2014). Lattice copies of c0and l∞in spaces of integrable functions for a vector measure. Dissertationes Mathematicae, 500, 1-68. doi:10.4064/dm500-0-