Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2017: 0.26

See all formats and pricing
More options …
Volume 64, Issue 6


Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals

Manjul Gupta / Antara Bhar
Published Online: 2015-01-10 | DOI: https://doi.org/10.2478/s12175-014-0287-6


In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M(X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M(X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.

MSC: Primary 46A45, 47B06, 47L20

Keywords: Lorentz sequence spaces; s-numbers of operators; Orlicz function and Orlicz sequence spaces; operator ideals

  • [1] ACHARYA, L. R.: Linear Operators and Approximation Quantites. Dissertation, Indian Institute of Technology, Kanpur, 2008. Google Scholar

  • [2] CARL, B.— STEPHANI, I.: Entropy, Compactness and the Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990. http://dx.doi.org/10.1017/CBO9780511897467CrossrefGoogle Scholar

  • [3] FORALWESKI, P.— HUDZIK, H.— SZYMASZKIEWICZ, L.: On some geometric and topological prpperties of generalized Orlicz-Lorentz sequence spaces, Math. Nachr. 281 (2008), 181–198.. http://dx.doi.org/10.1002/mana.200510594CrossrefWeb of ScienceGoogle Scholar

  • [4] GARLING, D. J. H.: On symmetric sequence spaces I, Proc. Lond. Math. Soc. (3) 16 (1966), 85–106. http://dx.doi.org/10.1112/plms/s3-16.1.85CrossrefGoogle Scholar

  • [5] GREGORY, D. A.: Vector-valued Sequence Spaces. Dissertation, Univ. of Michigan, Ann. Arbor, 1967. Google Scholar

  • [6] GROTHENDIECK, A.: Sur une notion de produit tensoriel topologique d’espaces vectoriels topologiques, et une, classes remarquable d’espaces liees a cette notion, C. R. Math. Acad. Sci. Paris 233 (1951), 1556–1558. Google Scholar

  • [7] GUPTA MANJUL— PRADHAN SHESADEV: On certain type of modular sequence spaces, Turkish J. Math. 32 (2008), 293–303. Google Scholar

  • [8] LINDENSTRAUSS, J.— TZAFRIRI, L.: Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1977. http://dx.doi.org/10.1007/978-3-642-66557-8CrossrefGoogle Scholar

  • [9] KAMTHAN, P. K.— GUPTA, M.: Sequence Spaces and Series. Lect. Notes Pure Appl. Math. 65, Marcel Dekker, Inc., New York-Basel, 1981. Google Scholar

  • [10] KATO, M.: On Lorentz Spaces l p,q{E}, HiroshimaMath. J. 6 (1976), 73–93. Google Scholar

  • [11] DE GRANDE-KIMPE, M.: Generalized sequence spaces, Bull. Soc. Math. Belgique XXIII (1971). Google Scholar

  • [12] KRASNOSELSKII, M. A.— RUTISKY, Y. B.: Convex Functions and Orlicz Spaces, Groningen, 1961. Google Scholar

  • [13] MUSIELAK, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. Google Scholar

  • [14] PATTERSON, J.: Generalized Sequence Spaces and Matrix Transformations. Dissertation, I.I.T. Kanpur, India, 1980. Google Scholar

  • [15] PIETSCH, A.: Zur Fredholmchen Theorie in lokalkonvexen Räumen, Studia Math. 22 (1963), 161–179. Google Scholar

  • [16] PIETSCH, A.: Einigeneue Klassen von kompakten linearen Abbildungen, Rev. Roumaine Math. Pures Appl. 8 (1963), 427–447. Google Scholar

  • [17] PIETSCH, A.: s-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223. Google Scholar

  • [18] PIETSCH, A.: Operator Ideals, Math. Monograph. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Google Scholar

  • [19] PIETSCH, A.: Eigenvalues and s-numbers, Cambridge Stud. Adv. Math. 13, Cambridge Univ. Press, Cambridge, 1987. Google Scholar

  • [20] RUCKLE, W.: The construction of sequence spaces that have schauder bases, Canad. J. Math. 19 (1967), 828–838. http://dx.doi.org/10.4153/CJM-1967-077-9CrossrefGoogle Scholar

  • [21] RUDIN, W.: Functional Analysis, McGraw-Hill, Inc., New York, 1976. Google Scholar

  • [22] SARGENT, W. L. C: Some sequence spaces related to the lp spaces, J. Lond. Math. Soc. (2) 35 (1960), 161–171. http://dx.doi.org/10.1112/jlms/s1-35.2.161CrossrefGoogle Scholar

  • [23] SCHMIDT, E.: Zur Theorie der linearen nichtlinearnen Integralgleichungen, Math. Ann. 63; 64 (1907), 433–476; 161–174. http://dx.doi.org/10.1007/BF01449770CrossrefGoogle Scholar

  • [24] YILMAZ, Y.— KERMAL ÖZDEMIR, M.— SOLAK, I.— CANDAN, M.: Operator on some vector-valued Orlicz sequence spaces, Fen Derg. 17 (2005), 59–71. Google Scholar

About the article

Published Online: 2015-01-10

Published in Print: 2014-12-01

Citation Information: Mathematica Slovaca, Volume 64, Issue 6, Pages 1475–1496, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-014-0287-6.

Export Citation

© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Manjul Gupta and Antara Bhar
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2014, Volume 108, Number 2, Page 733

Comments (0)

Please log in or register to comment.
Log in