Article contents
Banach spaces for which the space of operators has 2𝔠 closed ideals
Published online by Cambridge University Press: 19 March 2021
Abstract
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
- Type
- Analysis
- Information
- Creative Commons
- 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
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline7.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline8.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline9.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline10.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline11.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline12.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline13.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline14.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline15.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline16.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline17.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline18.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline19.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline20.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline21.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline22.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline23.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline24.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline25.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline26.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20210318173842215-0161:S2050509421000232:S2050509421000232_inline27.png?pub-status=live)
- 4
- Cited by