## Abstract and Applied Analysis

### The Banach-Saks Properties in Orlicz-Lorentz Spaces

#### Abstract

The Banach-Saks index of an Orlicz-Lorentz space ${\mathrm{\Lambda }}_{\phi ,w}(I)$ for both function and sequence case, is computed with respect to its Matuszewska-Orlicz indices of $\phi$. It is also shown that an Orlicz-Lorentz function space has weak Banach-Saks (resp., Banach-Saks) property if and only if it is separable (resp., reflexive).

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 423198, 8 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273239

Digital Object Identifier
doi:10.1155/2014/423198

Mathematical Reviews number (MathSciNet)
MR3182280

Zentralblatt MATH identifier
1292.46005

#### Citation

Kamińska, Anna; Lee, Han Ju. The Banach-Saks Properties in Orlicz-Lorentz Spaces. Abstr. Appl. Anal. 2014 (2014), Article ID 423198, 8 pages. doi:10.1155/2014/423198. https://projecteuclid.org/euclid.aaa/1412273239

#### References

• P. K. Lin, Köthe-Bochner Function Spaces, Birkhäuser, Boston, Mass, USA, 2004.
• J. Schreier, “Ein Gegenbeispiel zur theorie der schwachen konvergentz,” Studia Mathematica, vol. 2, pp. 58–62, 1930.
• A. Baernstein, II, “On reflexivity and summability,” Studia Mathematica, vol. 42, pp. 91–94, 1972.
• W. B. Johnson, “On quotients of ${L}_{p}$ which are quotients of ${l}_{p}$,” Compositio Mathematica, vol. 34, no. 1, pp. 69–89, 1977.
• J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, Germany, 1977.
• A. Kamínska and H. J. Lee, “Banach-Saks Properties of Musielak-Orlicz and Nakano sequence spaces,” Proceedings of the American Mathematical Society, vol. 142, no. 2, pp. 547–558, 2014.
• H. Knaust and E. Odell, “Weakly null sequences with upper ${l}_{p}$-estimates,” in Functional Analysis, vol. 1470 of Lecture Notes in Mathematics, pp. 85–107, Springer, Berlin, Germany, 1991.
• S. A. Argyros, G. Godefroy, and H. P. Rosenthal, “Descriptive set theory and Banach spaces,” in Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1007–1069, North-Holland, Amsterdam, The Netherlands, 2003.
• P. Cembranos, “The hereditary Dunford-Pettis property on $C(K,E)$,” Illinois Journal of Mathematics, vol. 31, no. 3, pp. 365–373, 1987.
• H. Knaust, “Orlicz sequence spaces of Banach-Saks type,” Archiv der Mathematik, vol. 59, no. 6, pp. 562–565, 1992.
• S. A. Rakov, “The Banach-Saks property for a Banach space,” Matematicheskie Zametki, vol. 26, no. 6, pp. 823–834, 1979.
• J. Cerdà, H. Hudzik, A. Kamińska, and M. Mastyło, “Geometric properties of symmetric spaces with applications to Orlicz-Lorentz spaces,” Positivity, vol. 2, no. 4, pp. 311–337, 1998.
• H. Hudzik, A. Kamińska, and M. Mastyło, “Geometric properties of some Calderón-Lozanovskiĭ spaces and Orlicz-Lorentz spaces,” Houston Journal of Mathematics, vol. 22, no. 3, pp. 639–663, 1996.
• H. Hudzik, A. Kamińska, and M. Mastyło, “On the dual of Orlicz-Lorentz space,” Proceedings of the American Mathematical Society, vol. 130, no. 6, pp. 1645–1654, 2002.
• A. Kamińska, “Some remarks on Orlicz-Lorentz spaces,” Mathematische Nachrichten, vol. 147, pp. 29–38, 1990.
• A. Kamińska and L. Maligranda, “Order convexity and concavity of Lorentz spaces ${\Lambda }_{p,w}, 0<p<\infty$,” Studia Mathematica, vol. 160, no. 3, pp. 267–286, 2004.
• A. Kamińska, L. Maligranda, and L. E. Persson, “Indices and regularizations of measurable functions,” in Function Spaces, vol. 213 of Lecture Notes in Pure and Applied Mathematics, pp. 231–246, Marcel Dekker, New York, NY, USA, 2000.
• W. Matuszewska and W. Orlicz, “On certain properties of $\phi$-functions,” Bulletin de l'Académie Polonaise des Sciences, vol. 8, pp. 439–443, 1960.
• A. Kamińska and Y. Raynaud, “Isomorphic copies in the lattice ${E}^{(\ast\,\!)}$ with applications to Orlicz-Lorentz spaces,” Journal of Functional Analysis, vol. 257, no. 1, pp. 271–331, 2009.
• E. Katirtzoglou, “Type and cotype of Musielak-Orlicz sequence spaces,” Journal of Mathematical Analysis and Applications, vol. 226, no. 2, pp. 431–455, 1998.
• A. Kamińska and B. Turett, “Type and cotype in Musielak-Orlicz spaces,” in Geometry of Banach Spaces, vol. 158 of London Mathematical Society Lecture Note Series, pp. 165–180, Cambridge Univeristy Press, Cambridge, UK, 1990.
• A. Kamińska, “Indices, convexity and concavity in Musielak-Orlicz spaces,” Functiones et Approximatio Commentarii Mathematici, vol. 26, pp. 67–84, 1998.
• B. Cuartero and M. A. Triana, “$(p,q)$-convexity in quasi-Banach lattices and applications,” Studia Mathematica, vol. 84, no. 2, pp. 113–124, 1986.
• N. J. Kalton, “Convexity conditions for nonlocally convex lattices,” Glasgow Mathematical Journal, vol. 25, no. 2, pp. 141–152, 1984.
• A. Kamińska, L. Maligranda, and L. E. Persson, “Indices, convexity and concavity of Calderón-Lozanovskii spaces,” Mathematica Scandinavica, vol. 92, no. 1, pp. 141–160, 2003.
• J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin, Germany, 1979.
• S. A. Rakov, “The Banach-Saks exponent of some Banach spaces of sequences,” Matematicheskie Zametki, vol. 32, no. 5, pp. 613–625, 1982.
• E. M. Semënov and F. A. Sukochev, “The Banach-Saks index,” Sbornik Matematicheskiĭ, vol. 195, no. 2, pp. 263–285, 2004.
• P. G. Dodds, E. M. Semenov, and F. A. Sukochev, “The Banach-Saks property in rearrangement invariant spaces,” Studia Mathematica, vol. 162, no. 3, pp. 263–294, 2004.
• H. P. Rosenthal, “Weakly independent sequences and the Banach-Saks property,” in Proceedings of the Durham Symposium on the Relations between Infinite Dimensional Convexity, p. 26, Durham, UK, July 1975.
• B. Beauzamy, “Banach-Saks properties and spreading models,” Mathematica Scandinavica, vol. 44, no. 2, pp. 357–384, 1979.
• K. Cho and C. Lee, “Alternative signs averaging properties in Banach spaces,” Journal of Applied Mathematics and Computing, vol. 16, pp. 497–507, 2004.
• C. A. Aliprantis and O. Burkinshaw, Positive Operators, vol. 119, Academic Press, Orlando, Fla, USA, 1985. \endinput