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Banach spaces for which the space of operators has 2𝔠 closed ideals

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

Published online by Cambridge University Press:  19 March 2021

Daniel Freeman ,
Thomas Schlumprecht Open the ORCID record for Thomas Schlumprecht [Opens in a new window]  and
András Zsák
Daniel Freeman
Affiliation:
Department of Mathematics and Statistics, St. Louis University, St. Louis, MO63103, USA; E-mail: daniel.freeman@slu.edu
Thomas Schlumprecht
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX77843, USA; E-mail: t-schlumprecht@tamu.edu Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 166 27, Prague, Czech Republic
András Zsák
Affiliation:
Peterhouse, University of Cambridge, Cambridge, CB2 1RD, United Kingdom; E-mail: a.zsak@dpmms.cam.ac.uk

Abstract

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We formulate general conditions which imply that ${\mathcal L}(X,Y)$, the space of operators from a Banach space X to a Banach space Y, has $2^{{\mathfrak {c}}}$ closed ideals, where ${\mathfrak {c}}$ is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in ${\mathcal L}\left (\ell _p\oplus \ell _q\right )$ is exactly $2^{{\mathfrak {c}}}$ for all $1<p<q<\infty $.

MSC classification

Primary: 47L20: Operator ideals
Secondary: 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.)
Type
Analysis
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e27
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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