Abstract
We characterize Cesàro–Orlicz function spaces \(Ces_{\varphi }\) containing order isomorphically isometric copy of \(l^{\infty }\) under some mild assumption imposed on the Orlicz function \(\varphi \). We discuss also some useful applicable sufficient conditions for the existence of such a copy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Astashkin, S.V., Maligranda, L.: Cesaro function spaces fail the fixed point property. Proc. Am. Math. Soc. 136(12), 4289–4294 (2008)
Astashkin, S.V., Maligranda, L.: Structure of Cesàro function spaces. Indag. Math. (N.S.) 20(3), 329–379 (2009)
Astashkin, S.V., Maligranda, L.: Structure of Cesàro function spaces: a survey. Banach Center Publ. 102, 13–40 (2014)
Astashkin, S.V., Maligranda, L.: Structure of Rademacher subspaces in Cesàro type spaces. Studia Math. 226, 259–279 (2015). https://doi.org/10.4064/sm226-3-4
Bennet, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press Inc, Boston (1988)
Boyd, D.W.: Indices of function spaces and their relationship to interpolation. Can. J. Math. 21, 1245–1254 (1969)
Boyd, D.W.: Indices for the Orlicz spaces. Pac. J. Math. 38(2), 315–323 (1971)
Chen, S.: Geometry of Orlicz spaces. Disserationes Math. (Rozprawy Mat.) 356, 1–204 (1996)
Cui, Y., Hudzik, H., Petrot, N., Suantai, S., Szymaszkiewicz, A.: Basic topological and geometric properties of Cesà ro-Orlicz spaces. Proc. Indian Acad. Sci. (Math. Sci.) 115(4), 461–476 (2005)
Cui, Y., Jie, L., Płuciennik, R.: Local uniform nonsquarness in Cesàro sequence spaces. Comm. Math. Prace Mat. 37, 47–58 (1997)
Cui, Y., Meng, C., Płuciennik, R.: Banach-Saks property and property \(\beta \) in Cesàro sequence spaces. Southeast Asian Bull. Math. 24, 201–210 (2000)
Curbera, G.P., Ricker, W.J.: Abstract Cesàro spaces, integral representations. J. Math. Anal. Appl. 441(1), 25–44 (2016)
Curbera, G.P., Ricker, W.J.: Solid extensions of the Cesàro operator on \(l^{p}\) and \(c_{0},\) Integr. Equ. Oper. Theory 80(1), 61–77 (2014)
Curbera, G.P., Ricker, W.J.: A feature of averaging. Integr. Equ. Oper. Theory 76(3), 447–449 (2013)
Curbera, G.P., Ricker, W.J.: The weak Banach-Saks property for function spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Serie A. Matemáticas https://doi.org/10.1007/s13398-016-0317-z
Delgado, O., Soria, J.: Optimal domain for the Hardy operator. J. Funct. Anal. 244(1), 119–133 (2007)
Granero, A.S., Hudzik, H.: On some proximinal subspaces of modular spaces. Acta Math. Hungar. 85(1–2), 59–79 (1999)
Hudzik, H.: Banach lattices with order isometric copies of \(l^{\infty }\). Indag. Math. (N.S.) 9(4), 521–527 (1998)
Kamińska, A., Kubiak, D.: On isometric copies of \( l_{\infty }\) and James constant in Cesàro–Orlicz sequence spaces. J. Math. Anal. Appl. 372, 574–584 (2010)
Kamińska, A., Maligranda, M., Persson, L.-E.: Indices and regularization of measurable functions. In: Function Spaces, The 5th Conference: Proceedings of the Conference at Poznań, Poland, pp. 231–246 (2000)
Kiwerski, T., Kolwicz, P.: Isomorphic copies of \( l^{\infty }\) in Cesàro–Orlicz function spaces. Positivity 21(3), 1015–1030 (2017). https://doi.org/10.1007/s11117-016-0449-6
Kiwerski, T., Tomaszewski, J.: Local approach to order continuity in Cesàro function spaces. J. Math. Anal. Appl. 455(2), 1636–1654 (2017). https://doi.org/10.1016/j.jmaa.2017.06.061
Kubiak, D.: A note on Cesàro–Orlicz sequence spaces. J. Math. Anal. Appl. 349, 291–296 (2009). https://doi.org/10.1016/j.jmaa.2008.08.022
Kufner, A., Maligranda, L., Persson, L.E.: The Hardy Inequality. About Its History and Some Related Results. Vydavatelsky Servis Publishing House, Pilsen (2007)
Krasnosel’skiĭ, M.A., Rutickiĭ, YaB: Convex functions and Orlicz Spaces. P. Noorddhoff Ltd., Groningen (1961). (translation)
Krein, S.G., Petunin, YuI, Semenov, E.M.: Interpolation of Linear Operators. Nauka, Moscow (1978)
Leśnik, K., Maligranda, M.: On abstract Cesàro spaces. I. Duality. J. Math. Anal. Appl. 424(2), 932–951 (2015). https://doi.org/10.1016/j.jmaa.2014.11.023
Leśnik, K., Maligranda, M.: On abstract Cesàro spaces. II. Optimal range. Integr. Equ. Oper. Theory 81(2), 227–235 (2015)
Leśnik, K., Maligranda, M.: Interpolation of abstract Cesàro, Copson and Tandori spaces. Indag. Math. (N.S.) 27(3), 764–785 (2016). https://doi.org/10.1016/j.indag.2016.01.009
Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Function Spaces. Springer, Berlin (1979)
Lozanovskiĭ, G.Ya.: On isomorphic Banach structures. Sibirsk. Math. Zh. 10, 93–98 (1969)
Maligranda, L.: Indices and interpolation. Disserationes Math. (Rozprawy Mat.) 234, 1–49 (1985)
Maligranda, L.: Orlicz Spaces and Interpolation, Sem. Mat., vol. 5. University of Campinas, Campinas (1989)
Maligranda, L., Petrot, N., Suantai, S.: On the James constant and \(B\)-convexity of Cesàro and Cesàro–Orlicz sequences spaces. J. Math. Anal. Appl. 326(1), 312–331 (2007). https://doi.org/10.1016/j.jmaa.2006.02.085
Matuszewska, W., Orlicz, W.: On certain properties of \(\varphi \)-functions. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8, 439–443 (1960)
Musielak, J.: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
Acknowledgements
Paweł Kolwicz was supported by the Ministry of Science and Higher Education of Poland, Grant Number 04/43/DSPB/0089.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kiwerski, T., Kolwicz, P. Isometric Copies of \(\varvec{l^\infty }\) in Cesàro–Orlicz Function Spaces. Results Math 73, 76 (2018). https://doi.org/10.1007/s00025-018-0834-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0834-5