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On a second numerical index for Banach spaces

Published online by Cambridge University Press:  28 January 2019

Sun Kwang Kim
Department of Mathematics, Chungbuk National University, 1 Chungdae-ro, Seowon-Gu, Cheongju, Chungbuk 28644, Republic of Korea (
Han Ju Lee*
Department of Mathematics Education, Dongguk University-Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul04620, Republic of Korea (
Miguel Martín
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (;
Javier Merí
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (;
*Corresponding author.


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.

Research Article
Copyright © Royal Society of Edinburgh 2019

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


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