Isomorphic ℓ^{𝑝}-subspaces in Orlicz-Lorentz sequence spaces

Kamińska, Anna; Raynaud, Yves

doi:10.1090/S0002-9939-06-08266-9

Isomorphic $\ell ^p$-subspaces in Orlicz-Lorentz sequence spaces
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by Anna Kamińska and Yves Raynaud PDF

Proc. Amer. Math. Soc. 134 (2006), 2317-2327 Request permission

Abstract:

Given a decreasing weight $w$ and an Orlicz function $\varphi$ satisfying the $\Delta _2$-condition at zero, we show that the Orlicz-Lorentz sequence space $d(w,\varphi )$ contains an $(1+\epsilon )$-isomorphic copy of $\ell _p, 1\le p<\infty$, if and only if the Orlicz sequence space $\ell _\varphi$ does, that is, if $p\in [\alpha _\varphi , \beta _\varphi ]$, where $\alpha _\varphi$ and $\beta _\varphi$ are the Matuszewska-Orlicz lower and upper indices of $\varphi$, respectively. If $\varphi$ does not satisfy the $\Delta _2$-condition, then a similar result holds true for order continuous subspaces $d_0(w,\varphi )$ and $h_\varphi$ of $d(w,\varphi )$ and $\ell _\varphi$, respectively.

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