Lipschitz Operators Associated with Weakly $p$-Compact and Unconditionally $p$-Compact Sets

Year 2024, Volume: 10 Issue: 3, 530 - 541, 30.09.2024

Ayşegül Keten Çopur , Ramazan İnal

https://doi.org/10.28979/jarnas.1451630

Abstract

The study introduces different categories of Lipschitz operators linked with weakly $p$-compact and unconditionally $p$-compact sets. It explores some properties of these operator classes derived from linear operators associated with these sets and examines their interconnections. Additionally, it denotes that these classes are extensions of the related linear operators. Moreover, the study evaluates the concept of majorization by scrutinizing both newly obtained and pre-existing results and draws some conclusions based on these findings. The primary method used to obtain the results in the study is the linearization of Lipschitz operators through the Lipschitz-free space constructed over a pointed metric space.

Keywords

Lipschitz compact operator, Lipschitz $p$-compact operator, Lipschitz weakly $p$-compact operator, Lipschitz unconditionally $p$-compact operator, majorization

References

• A. Jimenez-Vargas, J. M. Sepulcre, M. Villegas-Vallecillos, Lipschitz compact operators, Journal of Mathematical Analysis and Applications 415 (2) (2014) 889-901.
• A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Memoirs of the American Mathematical Society 16 (1955).
• D. P. Sinha, A. K. Karn, Compact operators whose adjoints factor through subspaces of $l_p$, Studia Mathematica 150 (1) (2002) 17-33.
• J. M. Kim, Unconditionally $p$-null sequences and unconditionally $p$-compact operators, Studia Mathematica 224 (2) (2014) 133-142.
• J. M. Kim, The ideal of unconditionally $p$-compact operators, The Rocky Mountain Journal of Mathematics 47 (7) (2017) 2277-2293.
• J. M. Kim, The ideal of weakly $p$-nuclear operators and its injective and surjective hulls, Journal of the Korean Mathematical Society 56 (1) (2019) 225-237.
• J. M. Kim, The ideal of weakly $p$-compact operators and its approximation property for Banach spaces, Annales Academiae Scientiarum Fennicae Mathematica 45 (2) (2020) 863-876.
• A. Keten Çopur, A. Satar, Some results on the $p$-weak approximation property in Banach spaces, Fundamental Journal of Mathematics and Applications 5 (4) (2022) 234-239.
• D. Achour, E. Dahia, P. Turco, Lipschitz $p$-compact mappings, Monatshefte für Mathematik 189 (2019) 595-609.
• N. Weaver, Lipschitz algebras, World Scientific, Singapore, 1999.
• N. J. Kalton, Spaces of Lipschitz and Hölder functions and their applications, Collectanea Mathematica 55 (2) (2004) 171-217.
• M. G. Cabrera-Padilla, A. Jimenez-Vargas, A new approach on Lipschitz compact operators, Topology and its Applications 203 (2016) 22-31.
• J. Diestel, J. H. Fourie, J. Swart, The metric theory of tensor products: Grothendieck's resume revisited. American Mathematical Society, 2008.
• J. M. Delgado, C. Pineiro, E. Serrano,Operators whose adjoints are quasi $p$-nuclear, Studia Mathematica 197 (3) (2010) 291-304.
• B. A. Barnes, Majorization, range inclusion, and factorization for bounded linear operators, Proceedings of the American Mathematical Society 133 (1) (2005) 155-162.
• A. Sahraoui, Majorizimg Lipschitz operators, Master's Thesis Université Mohamed Boudiaf (2021) M'sila.
• W. J. Davis, T. Figiel, W. B. Johnson, A. Peıczynski, Factoring weakly compact operators, Journal of Functional Analysis 17 (3) (1974) 311-327.
• G. Godefroy, A survey on Lipschitz-free Banach spaces, Commentationes Mathematicae 55 (2) (2015) 89-118.
• F. Albiac, N. J. Kalton, Topics in Banach space theory, Springer, New York, 2006.