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Banach-Saks properties of Musielak-Orlicz and Nakano sequence spaces

Authors: Anna Kamińska and Han Ju Lee
Journal: Proc. Amer. Math. Soc. 142 (2014), 547-558
MSC (2010): Primary 46B03, 46B25, 46B45
Published electronically: October 24, 2013
MathSciNet review: 3133996
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Abstract: In this paper Banach-Saks properties of Musielak-Orlicz sequence space $ \ell _\Phi $ are studied. It is shown that $ \ell _\Phi $ has the weak Banach-Saks property if and only if it is separable. Moreover it is proved that in $ \ell _\Phi $ both Banach-Saks type $ p$-properties, $ (BS_p)$ and $ (S_p)$, are equivalent and that the Schur property and $ (BS_\infty )$ also coincide in these spaces. As applications, we give characterizations of the weak Banach-Saks property and the $ (BS_p)$ property in the Nakano sequence space $ \ell ^{(p_n)}$ and weighted Orlicz sequence space $ \ell ^\phi (w)$, in terms of the sequence $ (p_n)$, and the Orlicz function $ \phi $ and the weight sequence $ w$, respectively.

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Additional Information

Anna Kamińska
Affiliation: Department of Mathematics, University of Memphis, Memphis, Tennessee 38152

Han Ju Lee
Affiliation: Department of Mathematics Education, Dongguk University - Seoul, 100-715 Seoul, Republic of Korea

Keywords: Banach-Saks properties, Schur property, Musielak-Orlicz space, Nakano space, variable exponent space, weighted Orlicz space
Received by editor(s): November 14, 2011
Received by editor(s) in revised form: March 19, 2012
Published electronically: October 24, 2013
Additional Notes: The second author is the corresponding author. He was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (20121A1A1006869)
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society