Abstract
AbstractWe 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
$\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$
.
Publisher
Cambridge University Press (CUP)
Reference17 articles.
1. Commutators on 𝐿_{𝑝}, 1≤𝑝<∞
2. Maximal Ideals in Some Spaces of Bounded Linear Operators
3. Commutators on (Σlq)p
4. Commutators on ℓ∞
5. The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space;Laustsen;J. Operator Theory,2006