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The ideal of Lipschitz classical p-compact operators and its injective hull Cover

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The ideal of Lipschitz classical p-compact operators and its injective hull

Moroccan Journal of Pure and Applied Analysis

Volume 8 (2022): Issue 1 (January 2022)

By: Toufik Tiaiba and Dahmane Achour

Open Access

|Jan 2022

Abstract

We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

References

[1] D. Achour, E. Dahia and P. Turco, The Lipschitz injective hull of Lipschitz operator ideals and applications, Banach J. Math. Anal, 14 (2020), 1241–1257.10.1007/s43037-020-00060-3
Search in Google Scholar Back to article
[2] D. Achour, E. Dahia and P. Turco, Lipschitz p-compact mapping, Monatshefte für Mathematik, 189 (2019), 595–609.10.1007/s00605-018-1252-1
Search in Google Scholar Back to article
[3] D. Achour, P. Rueda, E. A. Sanchez-Perez and R. Yahi, Lipschitz operator ideals and the approximation property, J. Math. Anal. Appl., 436, 1 (2016), 217–236.10.1016/j.jmaa.2015.11.050
Search in Google Scholar Back to article
[4] R. F. Arens and J. Eells Jr, On embedding uniform and topological spaces, Pacific J. Math., 6 (1956), 397–403.10.2140/pjm.1956.6.397
Search in Google Scholar Back to article
[5] A. Belacel and D. Chen, Lipschitz (p, r, s)-integral operators and Lipschitz (p, r, s)-nuclear operators, J. Math. Anal. Appl., 461, 2 (2018), 1115–1137.10.1016/j.jmaa.2018.01.056
Search in Google Scholar Back to article
[6] Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis, vol. 1. Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc. Providence, RI, (2000).10.1090/coll/048
Search in Google Scholar Back to article
[7] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, New York (1984).10.1007/978-1-4612-5200-9
Search in Google Scholar Back to article
[8] J. D. Farmer and W.B. Johnson, Lipschitz p-summing operators, Proc. Amer. Math. Soc., 137, 9 (2009), 2989–2995.10.1090/S0002-9939-09-09865-7
Search in Google Scholar Back to article
[9] J. H. Fourie, Quasi- and pseudo Λ-compact operators and locally convex spaces, Tamkang J. Math., 14 (1983), 171–181.
Search in Google Scholar Back to article
[10] J.H. Fourie, Injective and surjective hulls of classical p-compact operators with application to unconditionally p-compact operators, Studia Math., 240 (2018), 147–159.10.4064/sm8669-3-2017
Search in Google Scholar Back to article
[11] J.H. Fourie and J. Swart, Banach ideal of p-compact operators, Manuscripta Math., 26 (1979), 349–362.10.1007/BF01170259
Search in Google Scholar Back to article
[12] G. Godefroy, A survey on Lipschitz-free Banach spaces, Commentationes Mathematicae, 55, 2 (2015), 89–118.10.14708/cm.v55i2.1104
Search in Google Scholar Back to article
[13] A. Jiménez-Vargas, J.M. Sepulcre and Moisés Villegas-Vallecillos, Lipschitz compact operators, J. Math. Anal. Appl., 415, 2 (2014), 889–901.10.1016/j.jmaa.2014.02.012
Search in Google Scholar Back to article
[14] J.M. Kim, Unconditionally p-null sequences and unconditionally p-compact operators, Studia Math., 224 (2014), 133–142.10.4064/sm224-2-2
Search in Google Scholar Back to article
[15] E. Oja, A remark on the approximation of p-compact operators by finite-rank operators, J. Math. Anal. Appl., 387, 2 (2012), 949–952.10.1016/j.jmaa.2011.09.055
Search in Google Scholar Back to article
[16] A. Pietsch, Operator Ideals, Deutsch. Verlag Wiss, Berlin (1978), North-Holland, Amsterdam-London-New York-Tokyo (1980).
Search in Google Scholar Back to article
[17] D. P. Sinha and A. K. Karn, Compact operators whose adjoints factor through subspaces of ℓ_p, Studia Math., 150 (2002), 17–33.10.4064/sm150-1-3
Search in Google Scholar Back to article
[18] T. Terzioğlu, A characterization of compact linear mappings, Arch. Math., (Basel) 22 (1971), 76-78.10.1007/BF01222542
Search in Google Scholar Back to article
[19] N. Weaver, Lipschitz Algebras, World Scientific Publishing Co, Singapore (1999).10.1142/4100
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/mjpaa-2022-0003 | Journal eISSN: 2351-8227

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Language: English

Page range: 28 - 43

Submitted on: Nov 6, 2020

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Accepted on: Apr 30, 2021

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Published on: Jan 13, 2022

Published by: Sciendo

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Classical p-compact operator,

Lipschitz classical p-compact operator,

Lipschitz operator ideal,

Lipschitz injective hull,

Lipschitz unconditionally quasi p-nuclear operator

Related subjects:

Differential equations and dynamical systems,

Probability and statistics,

Numerical and computational mathematics

© 2022 Toufik Tiaiba, Dahmane Achour, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 8 (2022): Issue 1 (January 2022)

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