Taiwanese Journal of Mathematics

COPIES OF $c_0$ AND $\ell_\infty$ INTO A REGULAR OPERATOR SPACE

Yongjin Li, Donghai Ji, and Qingying Bu

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Abstract

For an Orlicz function $\varphi$ and a Banach lattice $X$, let $\ell_\varphi$ denote the Orlicz sequence space associated to $\varphi$, ${\mathcal L}^r(\ell_\varphi, X)$ denote the space of regular operators from $\ell_\varphi$ to $X$, and ${\mathcal K}^r(\ell_\varphi, X)$ denote the linear span of positive compact operators from $\ell_\varphi$ to $X$. In this paper, we show that if $\varphi$ and its complementary function $\varphi^\ast$ satisfy the $\Delta_2$-condition, then (a) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $\ell_\infty$; and (b) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $c_0$ if and only if ${\mathcal L}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $c_0$ and each positive linear operator from $\ell_\varphi$ to $X$ is compact.

Article information

Source
Taiwanese J. Math., Volume 16, Number 1 (2012), 207-215.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406537

Digital Object Identifier
doi:10.11650/twjm/1500406537

Mathematical Reviews number (MathSciNet)
MR2887861

Zentralblatt MATH identifier
1247.46018

Subjects
Primary: 46B42: Banach lattices [See also 46A40, 46B40] 46B20: Geometry and structure of normed linear spaces

Keywords
Orlicz sequence space regular operator space copies of $c_0$ and $\ell_\infty$

Citation

Li, Yongjin; Ji, Donghai; Bu, Qingying. COPIES OF $c_0$ AND $\ell_\infty$ INTO A REGULAR OPERATOR SPACE. Taiwanese J. Math. 16 (2012), no. 1, 207--215. doi:10.11650/twjm/1500406537. https://projecteuclid.org/euclid.twjm/1500406537


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