Reflexivity for spaces of regular operators on Banach lattices

Li, Yongjin; Bu, Qingying

doi:10.1090/proc/16018

Reflexivity for spaces of regular operators on Banach lattices
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by Yongjin Li and Qingying Bu PDF

Proc. Amer. Math. Soc. 150 (2022), 4811-4818 Request permission

Abstract:

We prove that if Banach lattices $E$ and $F$ are reflexive and each positive linear operator from $E$ to $F$ is compact then ${\mathcal L}^r(E;F)$, the space of all regular linear operators from $E$ to $F$, is reflexive. Conversely, if $E^\ast$ or $F$ has the bounded regular approximation property then the reflexivity of ${\mathcal L}^r(E;F)$ implies that each positive linear operator from $E$ to $F$ is compact. Analogously we also study the reflexivity for the space of regular multilinear operators on Banach lattices.

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