On a second numerical index for Banach spaces

TL;DR

This paper introduces a second numerical index for real Banach spaces with non-trivial Lie algebra, exploring its properties, differences from the classical index, and specific results for Hilbert spaces and spaces with unconditional bases.

Contribution

It defines a new numerical index for Banach spaces, analyzes its properties, and characterizes Hilbert spaces as the unique spaces with index one among certain classes.

Findings

Hilbert spaces have second numerical index one

Hilbert spaces are unique with this property among spaces with unconditional bases

Application to Bishop-Phelps-Bollobás property for numerical radius

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\ldots 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-Bollob\'as property for numerical radius is given.