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arXiv · 2212.06500

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

Arpita Mal
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Abstract

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.

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Arpita Mal. 2022-12-13. On joint numerical radius of operators and joint numerical index of a Banach space. https://arxiv.org/abs/2212.06500

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