Building ideals of two-Lipschitz operators between mertic and Banach spaces

In this paper, we present and characterize the injective hull of a two-Lipschitz operator ideals and the definition of two-Lipschitz dual operator ideal. Also we introduce two methods for creating ideals of two-Lipschitz operators from a pair of Lipschitz operator ideals. Namely, Lipschitzization and factorization method. We show the closedness, the injectivity and the symmetry of these two-Lipschitz ideals according to the closedness, injectivity and symmetry of the corresponding Lipschitz operator ideals. Some illustrative examples are given.Comment: 20 page

Lipschitz integral operators represented by vector measures

[EN] In this paper we introduce the concept of Lipschitz Pietsch-p-integral mappings, (1≤p≤∞), between a metric space and a Banach space. We represent these mappings by an integral with respect to a vectormeasure defined on a suitable compact Hausdorff space, obtaining in this way a rich factorization theory through the classical Banach spaces C(K), L_p(μ,K) and L_∞(μ,K). Also we show that this type of operators fits in the theory of composition Banach Lipschitz operator ideals. For p=∞, we characterize the Lipschitz Pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. Finally, the relationship between these mappings and some well known Lipschitz operators is given.We would like to thank the referee for his/her careful reading and useful suggestions. Also, we acknowledge with thanks the support of the general direction of scientific research and technological development (DGRSDT), Algeria.Dahia, E.; Hamidi, K. (2021). Lipschitz integral operators represented by vector measures. Applied General Topology. 22(2):367-383. https://doi.org/10.4995/agt.2021.15061OJS367383222D. Achour, P. Rueda, E. A. Sánchez-Pérez and R. Yahi, Lipschitz operator ideals and the approximation property, J. Math. Anal. Appl. 436 (2016), 217-236. https://doi.org/10.1016/j.jmaa.2015.11.050R. F. Arens and J. Eels Jr., On embedding uniform and topological spaces, Pacific J. Math 6 (1956), 397-403. https://doi.org/10.2140/pjm.1956.6.397A. Belacel and D. Chen, Lipschitz (p,r,s)-integral operators and Lipschitz (p,r,s)-nuclear operators, J. Math. Anal. Appl. 461 (2018) 1115-1137. https://doi.org/10.1016/j.jmaa.2018.01.056Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, vol. 1, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000. https://doi.org/10.1090/coll/048M. G. Cabrera-Padilla and A. Jiménez-Vargas, Lipschitz Grothendieck-integral operators, Banach J. Math. Anal. 9, no. 4 (2015), 34-57. https://doi.org/10.15352/bjma/09-4-3C. S. Cardassi, Strictly p-integral and p-nuclear operators, in: Analyse harmonique: Groupe de travail sur les espaces de Banach invariants par translation, Exp. II, Publ. Math. Orsay, 1989.D. Chen and B. Zheng. Lipschitz p-integral operators and Lipschitz p-nuclear operators, Nonlinear Anal. 75 (2012), 5270-5282. https://doi.org/10.1016/j.na.2012.04.044R. Cilia and J. M. Gutiérrez, Asplund Operators and p-Integral Polynomials, Mediterr. J. Math. 10 (2013), 1435-1459. https://doi.org/10.1007/s00009-013-0250-8R. Cilia and J. M. Gutiérrez, Ideals of integral and r-factorable polynomials, Bol. Soc. Mat. Mexicana 14 (2008), 95-124.J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511526138J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys Monographs 15, American Mathematical Society, Providence RI, 1977. https://doi.org/10.1090/surv/015N. Dunford and J. T. Schwartz, Linear Operators, Part I:General Theory, J. Wiley & Sons, New York, 1988.J. D. Farmer and W. B. Johnson, Lipschitz p-summing operators, Proc. Amer. Math. Soc. 137, no. 9 (2009), 2989-2995. https://doi.org/10.1090/S0002-9939-09-09865-7G. Godefroy, A survey on Lipschitz-free Banach spaces, Commentationes Mathematicae 55, no. 2 (2015), 89-118. https://doi.org/10.14708/cm.v55i2.1104A. Jiménez-Vargas, J. M. Sepulcre and M. Villegas-Vallecillos, Lipschitz compact operators, J. Math. Anal. Appl. 415 (2014), 889-901. https://doi.org/10.1016/j.jmaa.2014.02.012D. R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157-165. https://doi.org/10.2140/pjm.1970.33.157S. Okada, W. J. Ricker and E. A. Sánchez-Pérez, Optimal domain and integral extension of operators acting in function spaces, Operator theory: Adv. Appl., vol. 180, Birkhauser, Basel, 2008. https://doi.org/10.1007/978-3-7643-8648-1A. Persson and A. Pietsch.p-nuklear und p-integrale Abbildungen in Banach räumen, Studia Math. 33 (1969), 19-62. https://doi.org/10.4064/sm-33-1-19-62N. Weaver, Lipschitz Algebras, World Scientific Publishing Co., Singapore, 1999. https://doi.org/10.1142/410

Domination spaces and factorization of linear and multilinear summing operators

[EN] It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, sigma)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p(1), ... , p(n))-dominated multilinear operators and dominated (p(1), ... , p(n); sigma)-continuous multilinear operators.Supported by the Ministerio de Economia y Competitividad (Spain) under Grant MTM2015-66823-C2-2. Supported by the Ministerio de Economia y Competitividad (Spain) under Grant MTM2012-36740-C02-02.Achour, D.; Dahia, E.; Rueda, P.; Sánchez Pérez, EA. (2016). Domination spaces and factorization of linear and multilinear summing operators. Quaestiones Mathematicae. 39(8):1071-1092. https://doi.org/10.2989/16073606.2016.1253627S1071109239

Lipschitz p-compact mappings

We introduce the notion of Lipschitz p-compact operators. We show that they can be seen as a natural extension of the linear p-compact operators of Sinha and Karn and we transfer some properties of the linear case into the Lipschitz setting. Also, we introduce the notions of Lipschitz-free p-compact operators and Lipschitz locally p-compact operators. We compare all these three notions and show different properties. Finally, we exhibit examples to show that these three notions are different.Fil: Achour, Dahmane. University of M’sila; ArgeliaFil: Dahia, Elhadj. University of M’sila; ArgeliaFil: Turco, Pablo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

The Lipschitz injective hull of Lipschitz operator ideals and applications

We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.Fil: Achour, Dahmane. University of M’sila; ArgeliaFil: Dahia, Elhadj. University of M’sila; ArgeliaFil: Turco, Pablo Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin