On the Crawford number attaining operators

TL;DR

This paper investigates the conditions under which Crawford number attaining operators are dense in Banach spaces, establishing new density results and properties related to the Bishop-Phelps-Bollobás theorem.

Contribution

It proves that Crawford number attaining operators are dense in Banach spaces with the RNP and explores their density among compact operators with a 1-unconditional basis.

Findings

Crawford number attaining operators are dense in Banach spaces with the RNP.

Density of Crawford number attaining operators among compact operators with a 1-unconditional basis.

A Bishop-Phelps-Bollobás type property for Crawford numbers in certain Banach spaces.

Abstract

We study the denseness of Crawford number attaining operators on Banach spaces. Mainly, we prove that if a Banach space has the RNP, then the set of Crawford number attaining operators is dense in the space of bounded linear operators. We also see among others that the set of Crawford number attaining operators may be dense in the space of all bounded linear operators while they do not coincide, by observing the case of compact operators when the Banach space has a 1-unconditional basis. Furthermore, we show a Bishop-Phelps-Bollob\'as type property for the Crawford number for certain Banach spaces, and we finally discuss some difficulties and possible problems on the topic.