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

Kamińska, Anna; Lee, Han Ju

doi:10.1090/S0002-9939-2013-11842-3

Banach-Saks properties of Musielak-Orlicz and Nakano sequence spaces
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by Anna Kamińska and Han Ju Lee PDF

Proc. Amer. Math. Soc. 142 (2014), 547-558 Request permission

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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