On isometric copies of \(\ell _{\infty }\) and James constants in Cesáro-Orlicz sequence spaces
type: Item
instance of: scholarly article
title: On isometric copies of \(\ell _{\infty }\) and James constants in Cesáro-Orlicz sequence spaces
zbMATH Open document ID: 1210.46015
author: Damian Kubiak
author: Anna Kamińska
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-09-17T00:00:00Z
review text: 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-\(\ell^{(1)}_n\).
reviewed by: Henryk Hudzik
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 5787110
zbMATH Keywords: Cesàro-Orlicz sequence space
zbMATH Keywords: isometric copy of \(\ell _{\infty }\)
zbMATH Keywords: James constant
zbMATH Keywords: \(B\)-convexity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.07.011
OpenAlex ID: W2076609429
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DOI: 10.1016/J.JMAA.2010.07.011
description: scientific article; zbMATH DE number 5787110