Limit Theorems for the Numerical Index
We improve on a limit theorem (see Martin et al. (2011), Th. 5.1) for numerical index n(⋅) for large classes of Banach spaces including vector valued ℓp-spaces and ℓp-sums of Banach spaces where 1≤ p <∞. We introduce two conditions on a Banach space X, a local characterization condition (LCC) and a global characterization condition (GCC). We prove that if a norm on X satisfies the (LCC), then n(X)=limmn(Xm). An analogous result, in which N will be replaced by a directed, infinite set S will be proved for X satisfying the (GCC). We also present examples of Banach spaces satisfying the above mentioned conditions.
© 2013 Elsevier Inc.
Aksoy, A.G. Lewicki, “Limit Theorems for Numerical Index” Journal of Mathematical Analysis and Applications, vol. 398, issue 1, p. 296-302, 2013.