The injective hull of ideals of weighted holomorphic mappings

Abstract

We study the injectivity of normed ideals of weighted holomorphic mappings. To be more precise, the concept of injective hull of normed weighted holomorphic ideals is introduced and characterized in terms of a domination property. The injective hulls of those ideals -- generated by the procedures of composition and dual -- are described and these descriptions are applied to some examples of such ideals. A characterization of the closed injective hull of an operator ideal in terms of an Ehrling-type inequality -- due to Jarchow and Pelczy\'nski-- is established for weighted holomorphic mappings.

Keywords

• Weighted holomorphic mapping
• injective hull
• domination theorem
• operator ideal
• Ehrling inequality

References

1. D. Achour, E. Dahia: Building Ideals of two-Lipschitz operators between metric and Banach spaces, Mediterr. J. Math., 22 (2025), Article ID: 33.
2. D. Achour, E. Dahia and P. Turco: The Lipschitz injective hull of Lipschitz operator ideals and applications, Banach J. Math. Anal., 14 (3) (2020), 1241–1257.
3. K. D. Bierstedt, W. H. Summers: Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A, 54 (1) (1993), 70–79.
4. J. Bonet: Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 116 (2022), Article ID: 184.
5. J. Bonet, P. Dománski and M. Lindström: Essential norm and weak compactness of composition operators on weighted spaces of analytic functions, Canad. Math. Bull., 42 (2) (1999), 139–148.
6. J. Bonet, P. Domanski and M. Lindström: Weakly compact composition operators on weighted vector–valued Banach spaces of analytic mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 26 (2001), 233–248.
7. J. Bonet, M. Friz: Weakly compact composition operators on locally convex spaces, Math. Nachr., 245 (2002), 26–44.
8. G. Botelho, J. Campos and J. Santos: Operator ideals related to absolutely summing and Cohen strongly summing operators, Pac. J. Math., 87 (1) (2017), 1–7.

9. G. Botelho, L. A. Torres: Injective polynomial ideals and the domination property, Results Math., 75 (2020), Article ID: 24.
10. M. G. Cabrera-Padilla, A. Jiménez-Vargas and A. Keten Çopur: Weighted holomorphic mappings associated with pcompact type sets, Bull. Malays. Math. Sci. Soc., 48 (2025), Article ID: 32.
11. M. G. Cabrera-Padilla, A. Jiménez-Vargas and D. Ruiz-Casternado: On composition ideals and dual ideals of bounded holomorphic mappings, Results Math., 78 (2023), Article ID: 103.
12. J. S. Cohen: Absolutely p-summing, p-nuclear operators and their conjugates, Math. Ann., 201 (1973), 177–200.
13. G. Ehrling: On a type of eigenvalue problems for certain elliptic differential operators, Math. Scand., 2 (1954), 267–285.
14. 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.
15. M. González, J. M. Gutiérrez: Injective factorization of holomorphic mappings, Proc. Amer. Math. Soc., 127 (6) (1999), 1715–1721.
16. M. Gupta, D. Baweja: Weighted spaces of holomorphic functions on Banach spaces and the approximation property, Extracta Math., 31 (2) (2016), 123–144.
17. H. Jarchow: Locally convex spaces, Teubner, Stuttgart (1981).
18. J. M. Kim: The injective and surjective hulls of the ideal of (p, q)-compact operators and their approximation properties, J. Math. Anal. Appl., 473 (1) (2019), 71–86.
19. J. M. Kim: Corrigendum to “The ideal of weakly p-nuclear operators and its injective and surjective hulls” [J. Korean Math. Soc., 56 (1) (2019), 225–237], J. Korean Math. Soc., 57 (4) (2020), 1053–1057.
20. A. Manzano, P. Rueda and E. A. Sánchez-Pérez: Closed injective ideals of multilinear operators, related measures and interpolation, Math. Nachr., 293 (3) (2020), 510–532.
21. J. Mujica: Linearization of bounded holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc., 324 (2) (1991), 867–887.
22. A. Pietsch: The ideal of p-compact operators and its maximal hull, Proc. Am. Math. Soc., 142 (2013), 519–530.
23. A. Pietsch: Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York (1980). Translated from German by the author.
24. I. Stephani: Injektive Operatorenideale über der gesamtheit aller Banachräume und ihre topologische Erzeugung, Stud. Math., 38, 105-124 (1970).
25. T. Tiaiba, D. Achour: The ideal of Lipschitz classical p-compact operators and its injective hull, Moroccan J. Pure and Appl. Anal., 8 (1) (2022), 28–43.
26. R. Yahi, D. Achour and E. Dahia: Lipschitz closed injective hull ideals and Lipschitz interpolative ideals, Quaest. Math., (2024), 1–17. DOI: 10.2989/16073606.2024.2421890