Lipschitz integral operators represented by vector measures

Authors

  • Elhadj Dahia Ecole Normale Supérieure de Bousaada
  • Khaled Hamidi University of Mohamed El-Bachir El-Ibrahimi ; University of M’sila

DOI:

https://doi.org/10.4995/agt.2021.15061

Keywords:

Lipschitz Pietsch-p-integral operators, Lipschitz strictly p-integral operators, vector measure representation

Abstract

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 vector
measure 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.

Downloads

Download data is not yet available.

Author Biographies

Elhadj Dahia, Ecole Normale Supérieure de Bousaada

Laboratoire de Mathématiques et Physique Appliquées

Khaled Hamidi, University of Mohamed El-Bachir El-Ibrahimi ; University of M’sila

Department of Mathematics ; Laboratoire d’Analyse Fonctionnelle et Géométrie des Espaces

References

D. 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.2015.11.050

R. 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.1956.6.397

A. 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.2018.01.056

Y. 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/048

M. 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/09-4-3

C. 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.2012.04.044

R. Cilia and J. M. Gutiérrez, Asplund Operators and p-Integral Polynomials, Mediterr. J. Math. 10 (2013), 1435-1459. https://doi.org/10.1007/s00009-013-0250-8

R. 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/CBO9780511526138

J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys Monographs 15, American Mathematical Society, Providence RI, 1977. https://doi.org/10.1090/surv/015

N. 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/S0002-9939-09-09865-7

G. Godefroy, A survey on Lipschitz-free Banach spaces, Commentationes Mathematicae 55, no. 2 (2015), 89-118. https://doi.org/10.14708/cm.v55i2.1104

A. 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.2014.02.012

D. R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157-165. https://doi.org/10.2140/pjm.1970.33.157

S. 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/10.1007/978-3-7643-8648-1

A. 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-33-1-19-62

N. Weaver, Lipschitz Algebras, World Scientific Publishing Co., Singapore, 1999. https://doi.org/10.1142/4100

Downloads

Published

2021-10-01

How to Cite

[1]
E. Dahia and K. Hamidi, “Lipschitz integral operators represented by vector measures”, Appl. Gen. Topol., vol. 22, no. 2, pp. 367–383, Oct. 2021.

Issue

Section

Articles