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On a second numerical index for Banach spaces

  • Sun Kwang Kim (a1), Han Ju Lee (a2), Miguel Martín (a3) and Javier Merí (a3)

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.

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

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On a second numerical index for Banach spaces

  • Sun Kwang Kim (a1), Han Ju Lee (a2), Miguel Martín (a3) and Javier Merí (a3)

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