Relationship between Sublinear Operators and their Subdifferentials for Certain Classes of Lipschitz Summability

Résumé

Let SB(X, Y ) be the set of all bounded sublinear operators from a Banach space X into a complete Banach lattice Y ;
which is a pointed convex cone not salient in Lip0(X, Y ). In this paper, we are interested in studying the relationship between T and its
subdifferential ∇T (the set of all bounded linear operators u : X -→ Y such that u(x) ≤ T (x) for all x in X); concerning certain
notions of Lipschitz summability. We also answer negatively a question posed previously concerning this type of relation in the linear
case. For this, we introduce and study a new concept of summability in the category of Lipschitz operators, which we call ”super
Lipschitz p-summing operators”. We prove some characterizations in terms of a domination theorem and some properties of this notion.

Keywords: Banach lattice, Lipschitz p-dominated operator, Lipschitz p-summing operator, p-summing operator, sublinear operator
MSC: Primary 46B25, 46T99; Secondary 47H99, 47L20

REFERENCES
[1] Achour, D., & Belacel, A. (2014). Domination and factorization theorems for positive strongly p-summing operators. Positivity, 18(4), 785-804.   Search in Google Scholar   Digital Object Identifier
[2] Achour, D., & Mezrag, L. (2004). Little Grothendieck's theorem for sublinear operators. Journal of mathematical analysis and applications, 296(2), 541-552.‏ Search in Google Scholar   Digital Object Identifier
[3] Blasco, O. (1986). A class of operators from a Banach lattice into a Banach space. Collectanea mathematica, 13-22.‏ Search in Google Scholar   Article view
[4] Chávez-Domínguez, J. A. (2011). Duality for Lipschitz p-summing operators. Journal of functional Analysis, 261(2), 387-407.‏ Search in Google Scholar   Digital Object Identifier
[5] Chávez-Domínguez, J. (2012). Lipschitz (𝑞, 𝑝)-mixing operators. Proceedings of the American Mathematical Society, 140(9), 3101-3115.‏ Search in Google Scholar   Digital Object Identifier
[6] Chen, D., & Zheng, B. (2011). Remarks on Lipschitz 𝑝-summing operators. Proceedings of the American Mathematical Society, 139(8), 2891-2898.‏ Search in Google Scholar   Digital Object Identifier
[7] Chen, D., & Zheng, B. (2012). Lipschitz p-integral operators and Lipschitz p-nuclear operators. Nonlinear Analysis: Theory, Methods & Applications, 75(13), 5270-5282.‏  Search in Google Scholar   Digital Object Identifier
[8] Cohen, J. S. (1973). Absolutely p-summing, p-nuclear operators and their conjugates. Mathematische Annalen, 201(3), 177-200.‏  Search in Google Scholar   Article View
[9] Defant, A., & Floret, K. (1992). Tensor norms and operator ideals. Elsevier..‏  Search in Google Scholar   Article view
[10] Diestel, J., Jarchow, H., & Tonge, A. (1995). Absolutely summing operators (No. 43). Cambridge university press.‏  Search in Google Scholar   Digital Object Identifier
[11] Farmer, J., & Johnson, W. (2009). Lipschitz 𝑝-summing operators. Proceedings of the American Mathematical Society, 137(9), 2989-2995.‏  Search in Google Scholar   Digital Object Identifier  MathSciNet
[12] Krasteva, L.: The p+-Absolutely Summing Operators and Their Connection with (b-o)-Linear Operators [in Russian].
Diplomnaya Rabota (Thesis), Leningrad Univ., Leningrad (1971)
[13] Linke, Y.E.: Linear operators without subdifferentials. Sibirskii Mathematicheskii Zhurnal, 32(3), 219-221 (1991) 
[14] Lindenstrauss, J., & Pełczyński, A. (1968). Absolutely summing operators in $ ℒ_ {p} $-spaces and their applications. Studia Mathematica, 29(3), 275-326.‏ Search in Google Scholar   Article view
[15] Lindenstrauss, J., & Tzafriri, L. (2013). Classical Banach spaces II: function spaces (Vol. 97). Springer Science & Business Media.‏ Search in Google Scholar Digital Object Identifier
[16] Meyer-Nieberg, P. (2012). Banach lattices. Springer Science & Business Media.‏ Search in Google Scholar   Book view
[17] Mezrag, L., & Tallab, A. (2017). On Lipschitz $\tau (p) $-summing operators. In Colloquium Mathematicum (Vol. 147, pp. 95-114). Instytut Matematyczny Polskiej Akademii Nauk.‏ Search in Google Scholar   Digital Object Identifier
[18] Pietsch, A. (1967). Absolut p-summierende Abbildungenin normierten Raumen. Studia Math., 28, 333-353.‏ Search in Google Scholar   Article view
[19] Saadi, K. (2015). Some properties of Lipschitz strongly p-summing operators. Journal of Mathematical Analysis and Applications, 423(2), 1410-1426.‏ Search in Google Scholar   Digital Object Identifier
[20] Yahi, R., Achour, D., & Rueda, P. (2016). Absolutely summing Lipschitz conjugates. Mediterranean Journal of Mathematics, 13, 1949-1961.‏ Search in Google Scholar   Digital Object Identifier

Communicated Editor: Berbiche Mohamed
Manuscript received Jan 17, 2024; revised Mar 27, 2024; accepted Apr 10, 2024; published May 13, 2024.