On the Crawford number attaining operators

TL;DR

This paper investigates the denseness of Crawford number attaining operators on Banach spaces. It proves that if $X$ has the Radon-Nikodým property, then $\mathop{\mathrm{CNA}}(X)$ is dense in $\mathcal{L}(X)$, and shows that $\mathop{\mathrm{CNA}}_\mathcal{K}(X)$ can be dense in $\mathcal{K}(X)$ under a $1$-unconditional basis even when $\mathop{\mathrm{CNA}}(X)$ and $\mathcal{K}(X)$ do not coincide; it also establishes a Bishop-Phelps-Bollobás type property for the Crawford number on spaces that are uniformly convex and uniformly smooth, yielding a weak BPBp result. The authors further explore the denseness of adjoint operators and structural properties ($\alpha$, $\beta$), and pose several open problems on adjoint denseness, BPBp extensions, and stability of denseness in the full operator space. Overall, the work delineates positive density results under geometric hypotheses while highlighting substantial challenges in extending these results to general 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-Bollobás type property for the Crawford number for certain Banach spaces, and we finally discuss some difficulties and possible problems on the topic.

Theorems & Definitions (25)

• Definition 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5