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THE MAXIMAL IDEAL IN THE SPACE OF OPERATORS ON $\boldsymbol {(\sum {\ell }_{q})_{c_{0}}}$

Part of: Linear spaces and algebras of operators Special classes of linear operators

Published online by Cambridge University Press:  21 February 2022

DIEGO CALLE CADAVID ,
MONIKA  and
BENTUO ZHENG Open the ORCID record for BENTUO ZHENG [Opens in a new window]
DIEGO CALLE CADAVID
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA e-mail: dcllcdvd@memphis.edu
MONIKA
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA e-mail: myadav@memphis.edu
BENTUO ZHENG*
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA
*
e-mail: bzheng@memphis.edu
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Abstract

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 $\mathcal {L}((\sum {\ell }_{q})_{c_{0}})$.

Keywords

unconditional basisstrictly singular operatorsmaximal ideal

MSC classification

Primary: 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.)
Secondary: 47L20: Operator ideals

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 340 - 348
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Bentuo Zheng’s research is supported in part by Simons Foundation Grant 585081.

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