On a second numerical index for Banach spaces. Issue 2 (April 2020)

• Record Type: Journal Article
• Authors: Kim, Sun Kwang; Lee, Han Ju; Martín, Miguel; Merí, Javier
• Is Part Of: Proceedings. Volume 150:Issue 2(2020)
• Page Start: 1003
• Page End: 1051
• Publication Date: 2020-04
• Subjects: Banach space, -- numerical range, -- numerical index, -- skew hermitian operator, -- Bishop-Phelps-Bollobás property for the numerical radius; 46B04, -- 46B20, -- 47A12; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2018.75
• Languages: English
• ISSNs: 0308-2105

Abstract

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.