On isometric copies of \(\ell _{\infty }\) and James constants in Cesáro-Orlicz sequence spaces
Define on the space \(\ell^0\) of all real sequences the averaging operator \(G:\ell^0\rightarrow\ell^0\) by \[ Gx(n)=\frac{1}{n}\overset{n}{\underset{i=1}{\sum}}\left|x(i)\right|,\;\;n\in\mathbb{N}. \] Given any Orlicz function \(\varphi\), the modular \[ I_{\text{ces}_\varphi}(x)=I_\varphi (Gx)=\overset{n}{\underset{i=1}{\sum}}\varphi(Gx(n)) \] 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-\(\ell^{(1)}_n\).
scientific article; zbMATH DE number 5787110
scientific article