THE MAXIMAL IDEAL IN THE SPACE OF OPERATORS ON $\boldsymbol {(\sum {\ell }_{q})_{c_{0}}}
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 i...
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| Veröffentlicht in: | Bulletin of the Australian Mathematical Society Jg. 106; H. 2; S. 340 - 348 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge, UK
Cambridge University Press
01.10.2022
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| Schlagworte: | |
| ISSN: | 0004-9727, 1755-1633 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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
$\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$
. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0004-9727 1755-1633 |
| DOI: | 10.1017/S0004972722000028 |