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On isometric copies of $\ell _{\infty }$ and James constants in Cesáro-Orlicz sequence spaces. (English) Zbl 1210.46015
Define on the space $\ell^0$ of all real sequences the averaging operator $G:\ell^0\rightarrow\ell^0$ by $$Gx(n)=\frac{1}{n}\overset{n}\to{\underset{i=1}\to{\sum}}\left|x(i)\right|,\;\;n\in\mathbb{N}.$$ Given any Orlicz function $\varphi$, the modular $$I_{\text{ces}_\varphi}(x)=I_\varphi (Gx)=\overset{n}\to{\underset{i=1}\to{\sum}}\varphi(Gx(n))$$ defined on $\ell^0$ is convex and defines the Cesàro-Orlicz sequence space $$\text{ces}_\varphi=\{x\in\ell^0: I_{\text{ces}_\varphi}(\lambda x)<\infty \text{ for some }\lambda>0\}$$ with the Luxemburg norm $$\left\|x\right\|_{\text{ces}_\varphi}=\inf\{\varepsilon>0:I_{\text{ces}_\varphi}(x\slash\varepsilon)\leq 1\}=\left\|Gx\right\|_\varphi.$$ First, it is proved that if $\varphi$ does not satisfy the $\delta_2$-condition (a growth condition in a neighbourhood of zero) and the Orlicz class $\{x: I_\varphi(x)<\infty\}$ is closed under the averaging operator $G$, then $\text{ces}_\varphi$ contains an order isometric copy of $\ell_\infty$. Next, tree conditions such that each of them is equivalent to the fact that the Orlicz class $\{x:I_\varphi(x)<\infty\}$ is closed under the averaging operator are presented. An example of an Orlicz function $\varphi$ such that the Orlicz class $\{x: I_\varphi(x)<\infty\}$ is not closed under the averaging operator and the space $\text{ces}_\varphi$ contains an order isometric copy of $\ell_\infty$ is given. Finally, it is proved that, for any natural $n\geq 2$, the James constants $J_n^s(\text{ces}_\varphi)$ are equal to $n$, which means that, for any natural $n\geq 2$, the Cesàro-Orlicz space is not uniformly non-$\ell^{(1)}_n$.

46B20Geometry and structure of normed linear spaces
46B45Banach sequence spaces
Full Text: DOI
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