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Published online by Cambridge University Press: 21 February 2022
We study the isomorphic structure of $(\sum {\ell }_{q})_{c_{0}}\ (1< q<\infty )$ and prove that these spaces are complementably homogeneous. We also show that for any operator T from
$(\sum {\ell }_{q})_{c_{0}}$ into
${\ell }_{q}$, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$ that is isometric to
$(\sum {\ell }_{q})_{c_{0}}$ and the restriction of T on X has small norm. If T is a bounded linear operator on
$(\sum {\ell }_{q})_{c_{0}}$ which is
$(\sum {\ell }_{q})_{c_{0}}$-strictly singular, then for any
$\epsilon>0$, there is a subspace X of
$(\sum {\ell }_{q})_{c_{0}}$ which is isometric to
$(\sum {\ell }_{q})_{c_{0}}$ with
$\|T|_{X}\|<\epsilon $. As an application, we show that the set of all
$(\sum {\ell }_{q})_{c_{0}}$-strictly singular operators on
$(\sum {\ell }_{q})_{c_{0}}$ forms the unique maximal ideal of
$\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$.
Bentuo Zheng’s research is supported in part by Simons Foundation Grant 585081.