Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-fnprw Total loading time: 0.209 Render date: 2022-08-15T05:17:39.847Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

Published online by Cambridge University Press:  02 February 2006

Yun Sung Choi
Affiliation:
Department of Mathematics, POSTECH, Pohang 790-784, South Korea (mathchoi@euclid.postech.ac.kr)
Domingo Garcia
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot, Valencia, Spain (domingo.garcia@uv.es)
Sung Guen Kim
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea (sgk317@knu.ac.kr)
Manuel Maestre
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot, Valencia, Spain (manuel.maestre@uv.es)
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:

(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.

(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.

(iii) The inequalities

$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$

for every Banach space $E$.

(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.

(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.

(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.

Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2006
You have Access
19
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.

THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE
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.

THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE
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.

THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE
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? *