Abstract
We introduce and study the new ideal of strongly Lorentz summing operators between Banach spaces generated by strongly Lorentz sequence spaces to study the adjoints of the Lorentz summing operators. We also prove the related dual result: an operator is Lorentz summing if and only if its adjoint is strongly Lorentz summing. Some examples, counterexamples and connections with the theory of absolutely summing operators are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Attallah, A., Achour, D.: Vector valued Lorentz sequence spaces and their dual spaces. J. Nonlinear Appl. Sci. 10, 338–353 (2017)
Apiola, H.: Duality between spaces of \(p\)-summable sequences, \((p, q)\)-summing operators and characterizations of nuclearity. Math. Ann 219, 53–64 (1976)
Botelho, G., Campos, J.R.: On the transformation of vector-valued sequences by multilinear operators. Monatsh. Math 183, 415–435 (2017)
Cohen, J.: Absolutely \(p\)-summing, \(p\)-nuclear operators and their conjugates, conjugates. Math. Ann 201, 177–200 (1973)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)
Grothendieck, A.: Resume de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo 8, 1–79 (1953)
Kato, M.: Conjugates of \((p, q;r)\)-absolutely summing operators. Hiroshima Math. J. 5, 127–134 (1975)
Kato, M.: On Lorentz spaces \(\ell _{p, q}\left( E\right)\). Hiroshima Math. J. 6, 73–93 (1976)
Lindenstrauss, J., Pełczynski, A.: Absolutely summing operators in \({\cal{L}}_{p}\) spaces and their applications. Stud. Math. 29, 275–326 (1968)
Mastyło, M.: An operator ideal generated by Orlicz spaces. Math. Ann. 376, 1675–1703 (2020)
Matos, M.C., Pellegrino, D.: Lorentz summing mappings. Math Nachr. 283(10), 1409–1427 (2010)
Miyazaki, K.: \((p, q;r)\)-absolutely summing operators. J. Math Soc. Jpn. 24, 341–354 (1972)
Miyazaki, K.: \((p, q)\)-nuclear and \((p, q)\)-integral operators. Hiroshima Math. J. 4, 99–132 (1974)
Miyazaki, K.: Some remarks on intermediate spaces. Bull. Kyushu Inst. Technol. 15, 1–23 (1968)
Pietsch, A.: Eigenvalues and S-Numbers . Cambridge University Press, Cambridge (1987)
Pietsch, A.: Operator Ideals. Deutsch. Verlag Wiss., Berlin, 1978; North-Holland, Amsterdam–London–New York-Tokyo, 1980.
Pietsch, A.: Absolute \(p\)-summierende Abbildungen in normierten Raumen. Stud. Math 28, 333–353 (1967)
Tomczak, N.: A remark on \((s;t)\)-absolutely summing operators in \({\cal{L}}_{p}\)-spaces. Stud. Math 35, 97–100 (1970)
Acknowledgements
We would like to thank the referee for his valuable comments and suggestions. Also, we acknowledge with thanks the support of the General Direction of Scientific Research and Technological Development (DGRSDT), Algeria.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anna Kaminska.
Rights and permissions
About this article
Cite this article
Achour, D., Attallah, A. Operator ideals generated by strongly Lorentz sequence spaces. Adv. Oper. Theory 6, 35 (2021). https://doi.org/10.1007/s43036-021-00132-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-021-00132-7