Banach spaces with polynomial numerical index 1

TL;DR

This paper characterizes Banach spaces with polynomial and holomorphic numerical index 1 under the Radon-Nikodym property, providing new insights into their structure and properties.

Contribution

It introduces the holomorphic numerical index and characterizes Banach spaces with this property when they possess the Radon-Nikodym property.

Findings

Banach spaces with polynomial numerical index 1 are characterized under the Radon-Nikodym property.

The holomorphic numerical index is introduced and characterized for spaces with the Radon-Nikodym property.

New relationships between numerical indices and geometric properties of Banach spaces are established.

Abstract

We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikod\'ym property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikod\'ym property.