On joint numerical radius of operators and joint numerical index of a Banach space

Generalizing the notion of numerical range and numerical radius of an operator on a Banach space, we introduce the notion of joint numerical range and joint numerical radius of tuple of operators on a Banach space. We study the convexity of the joint numerical range. We show that the joint numerical...

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Bibliographic Details
Published inarXiv.org
Main Author Mal, Arpita
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 13.12.2022
Subjects
Banach spaces
Convexity
Mathematics - Functional Analysis
Operators
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Summary:Generalizing the notion of numerical range and numerical radius of an operator on a Banach space, we introduce the notion of joint numerical range and joint numerical radius of tuple of operators on a Banach space. We study the convexity of the joint numerical range. We show that the joint numerical radius defines a norm if and only if the numerical radius defines a norm on the corresponding space. Then we prove that on a finite-dimensional Banach space, the joint numerical radius can be retrieved from the extreme points. Furthermore, we introduce a notion of joint numerical index of a Banach space. We explore the same for direct sum of Banach spaces. Applying these results, finally we compute the joint numerical index of some classical Banach spaces.
ISSN:2331-8422
DOI:10.48550/arxiv.2212.06500