## 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