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Banach-Saks property and property \((\beta)\) in Cesàro sequence spaces. (English) Zbl 0956.46003

Remind that a real Banach space \(X\) is said to have the Banach-Saks (resp.weak Banach-Saks) property if every bounded (resp. weak null) sequence \(\{x_n\}\) in \(X\) admits a subsequence \(\{z_n\}\) such that the sequence of its arithmetic means \(\{1/n(z_1+z_2+\cdots+z_n+\cdots)\}\) is convergent in norm of \(X\). \(X\) is said to have property \(\beta\) if and only if, for every \(\varepsilon>0\), there exists \(\delta>0\) such that, for each \(x\) of the unit ball \(B(X)\) and for each sequence \(\{x_n\}\subset B(X)\) with \(\|x_n-x_m\|\geq \varepsilon,\;n\neq m,\) there is an index \(k\) such that \(\mathbb\|\frac{x+x_k}{2}\mathbb\|\leq 1-\delta\). Remind as well that the Cesàro sequence space denoted by \(ces_p\;(1<p<\infty)\) consists of all real sequences \(x=\{x_n\}\) such that \(\|x\|_{ces_p}:=(\sum_{n=1}^{\infty}\mathbb(\frac{1}{n}\sum_{i=1}^n |x_i|\mathbb)^p)^{1/p} <\infty\). The authors prove that \(ces_p\) has property \(\beta\) thus both of the spaces \(ces_p\) and \((ces_p)^*\) have the Banach-Saks property. A new constant \(C(X)\) is introduced for any Banach space \(X\) as follows: \(C(X)=\sup \{A(\{x_n\}):\{x_n\}\) is a weakly null sequence, \(\|x_n\|=1\), \(n\geq 1\}\) where \(A(\{x_n\})=\lim_{n\rightarrow\infty}\inf \{\|x_i+x_j\|:i,j\geq n,\;i\neq j\}\). The authors show that the inequality \(C(X)<2\) implies fulfilling of the weak Banach-Saks property for \(X\). Moreover they calculate \(C(ces_p)=2^{1/p}\).

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46B20 Geometry and structure of normed linear spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
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