Hostname: page-component-f554764f5-qhdkw Total loading time: 0 Render date: 2025-04-11T06:44:41.292Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Special classes of linear operators Linear spaces and algebras of 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: [email protected]
MONIKA
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA e-mail: [email protected]
BENTUO ZHENG*
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152-3240, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

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

References

Albiac, F. and Kalton, N., Topics in Banach Space Theory, 2nd edn, Graduate Texts in Mathematics, 233 (Springer, New York, 2016).CrossRefGoogle Scholar
Chen, D., Johnson, W. and Zheng, B., ‘Commutators on ${(\sum {\ell}_q)}_{\ell_p}$ ’, Studia Math. 206(2) (2011), 175–190.CrossRefGoogle Scholar
Dosev, D. and Johnson, W., ‘Commutators on ${\ell}_{\infty }$ ’, Bull. Lond. Math. Soc. 42(1) (2010), 155–169.CrossRefGoogle Scholar
Dosev, D., Johnson, W. and Schechtman, G., ‘Commutators ${L}_p$ , $1\le p<\infty$ ’, J. Amer. Math. Soc. 26(1) (2013), 101–127.CrossRefGoogle Scholar
Gohberg, I., Markus, A. and Feldman, I., ‘Normally solvable operators and ideals associated with them’, Bul. Acad. Ştiinţe Repub. Mold. Mat. 10(76) (1960), 51–70; English translation, Amer. Math. Soc. Transl. 61 (1967), 63–84.Google Scholar
James, R., ‘Uniformly non-square Banach spaces’, Ann. of Math. (2) 80 (1964), 542–550.CrossRefGoogle Scholar
Kaminska, A., Popov, A., Spinu, E., Tcaciuc, A. and Troitsky, V., ‘Norm closed operator ideals in Lorentz sequence spaces’, J. Math. Anal. Appl. 389(1) (2012), 247–260.CrossRefGoogle Scholar
Kania, T. and Laustsen, N., ‘Uniqueness of the maximal ideal of operators on the ${\ell}_p$ -sum of ${\ell}_{\infty}^n\left(n\in \mathbb{N}\right)$ for $1<p<\infty$ ’, Math. Proc. Cambridge Philos. Soc. 160(3) 2016, 413–421.CrossRefGoogle Scholar
Krivine, J. and Maurey, B., ‘Stable Banach spaces’, Israel J. Math. 39(4) (1981), 273–295.CrossRefGoogle Scholar
Laustsen, N., Loy, R. and Read, C., ‘The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces’, J. Funct. Anal. 214 (2004), 106–131.CrossRefGoogle Scholar
Laustsen, N., Odell, E., Schlumprecht, T. and Zsak, A., ‘Dichotomy theorems for random matrices and closed ideals of operators on ${({\oplus}_{n=1}^{\infty }{\ell}_1^n)}_{c_0}$ ’, J. Lond. Math. Soc. (2) 86 (2012), 235–258.CrossRefGoogle Scholar
Laustsen, N., Schlumprecht, T. and Zsak, A., ‘The lattice of closed ideals in the Banach algebra of operators on a certain dual Banach space’, J. Operator Theory 56 (2006), 391–402.Google Scholar
Leung, D., ‘Maximal ideals in some spaces of bounded linear operators’, Proc. Edinb. Math. Soc. (2) 61(1) (2018), 251–264.CrossRefGoogle Scholar
Lin, P., Sari, B. and Zheng, B., ‘Norm closed ideals in the algebra of bounded linear operators on Orlicz sequence spaces’, Proc. Amer. Math. Soc. 142 (2014), 1669–1680.CrossRefGoogle Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces (Springer, Berlin–Heidelberg, 2006).Google Scholar
Zheng, B., ‘ Commutators on $\mathbf{\mathcal{L}}({(\sum {\ell}_q)}_{\ell_1})$ ’, J. Math. Anal. Appl. 413 (2014), 284–290.CrossRefGoogle Scholar
Zheng, B., ‘ Operators on ${L}_p(2<p<\infty )$ which factor through ${X}_p$ ’, J. Funct. Anal. 273(2) (2014), 782–798.CrossRefGoogle Scholar