Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-54jdg Total loading time: 0.315 Render date: 2022-08-08T01:36:56.550Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

Published online by Cambridge University Press:  09 July 2021

Sheldon Dantas
Affiliation:
Departament de Matemàtiques and Institut Universitari de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Campus del Riu Sec. s/n, 12071 Castelló Spain (dantas@uji.es)
Mingu Jung*
Affiliation:
Basic Science Research Institute and Department of Mathematics, POSTECH, Pohang790-784, Republic of Korea
Gonzalo Martínez-Cervantes
Affiliation:
Universidad de Murcia, Departamento de Matemáticas, Campus de Espinardo 30100Murcia, Spain (gonzalo.martinez2@um.es)

Abstract

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent:

  1. (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ;

  2. (ii) Every operator from E into F attains its norm;

  3. (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ ,

where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

Type
Research Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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.)
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *