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On a second numerical index for Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators

Published online by Cambridge University Press:  28 January 2019

Sun Kwang Kim ,
Han Ju Lee ,
Miguel Martín  and
Javier Merí
Sun Kwang Kim
Affiliation:
Department of Mathematics, Chungbuk National University, 1 Chungdae-ro, Seowon-Gu, Cheongju, Chungbuk 28644, Republic of Korea (skk@chungbuk.ac.kr)
Han Ju Lee*
Affiliation:
Department of Mathematics Education, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul04620, Republic of Korea (hanjulee@dongguk.edu)
Miguel Martín
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (mmartins@ugr.es; jmeri@ugr.es)
Javier Merí
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (mmartins@ugr.es; jmeri@ugr.es)
*
*Corresponding author.
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Abstract

We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

Keywords

Banach spacenumerical rangenumerical indexskew hermitian operatorBishop-Phelps-Bollobás property for the numerical radius

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces 47A12: Numerical range, numerical radius
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 1003 - 1051
Copyright
Copyright © Royal Society of Edinburgh 2019

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Footnotes

Dedicated to Rafael Payá on the occasion of his 60th birthday

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