Norm-attaining operators which satisfy a Bollobás type theorem (Q2662213)

This paper deals with a property related to the the Bishop-Phelps-Bollobás property (BPBp in short) introduced by \textit{M. D. Acosta} et al. [J. Funct. Anal. 254, No. 11, 2780--2799 (2008; Zbl 1152.46006)]. The BPBp of a pair \((X,Y)\) is an improved version of the denseness of the set of norm attaining (bounded linear) operators from \(X\) to \(Y\), in the same way that \textit{B. Bollobás} [Bull. Lond. Math. Soc. 2, 181--182 (1970; Zbl 0217.45104)] improved the classical Bishop-Phelps theorem [\textit{E. Bishop} and \textit{R. R. Phelps}, Bull. Am. Math. Soc. 67, 97--98 (1961; Zbl 0098.07905)]. In the last few years, a number of properties related to the BPBp have appeared in the journal literature. One of them is the \textit{\(\mathrm{L}_{o,o}\)} property of a pair \((X,Y)\) of Banach spaces [\textit{S. Dantas}, Math. Nachr. 290, No. 5--6, 774--784 (2017; Zbl 1372.46009); \textit{S. Dantas} et al., J. Math. Anal. Appl. 468, No. 1, 304--323 (2018; Zbl 1412.46022)] which is defined as follows: for every norm-one (linear) operator \(T\colon X\longrightarrow Y\) and every \(\varepsilon>0\), there is \(\eta(\varepsilon,T)>0\) such that whenever \(x\in X\) with \(\|x\|=1\) satisfies \(\|Tx\|>1-\eta(\varepsilon,T)\), there is \(y\in X\) with \(\|y\|=1\) such that \(\|Ty\|=1\) and \(\|x-y\|<\varepsilon\). Two observations are pertinent. First, the property stronger than \(\mathrm{L}_{o,o}\) that is defined requiring that \(\eta\) only depends on \(\varepsilon\) and not on the operator is not possible if both spaces \(X\) and \(Y\) have dimension greater than or equal to two [\textit{S. Dantas} et al., Linear Multilinear Algebra 68, No. 9, 1767--1778 (2020; Zbl 1465.46011)]. Secondly, when a pair \((X,Y)\) has the \(\textrm{L}_{o,o}\), then all bounded linear operator from \(X\) to \(Y\) attain their norm, and so \(X\) has to be reflexive thanks to James's theorem. The paper under review is dedicated to the study the set \(\mathcal{A}_{\|\cdot\|}(X,Y)\) of those norm-one operators \(T\) from \(X\) to \(Y\) for which there is \(\eta(\varepsilon,T)>0\) for every \(\varepsilon>0\) satisfying the requirements of \(\mathrm{L}_{o,o}\). Of course, \((X,Y)\) has \(\mathrm{L}_{o,o}\) if and only if \(\mathcal{A}_{\|\cdot\|}(X,Y)\) coincides with the set of all norm-one operators from \(X\) to \(Y\). But the study of this set by itself could be interesting, especially in the case when \(X\) is not reflexive. As a sample of the results presented in the paper, we mention the following ones: every norm-attaining norm-one functional on \(c_0\) belongs to \(\mathcal{A}_{\|\cdot\|}(c_0,\mathbb{K})\), a result which is false for \(\ell_1\) and \(\ell_\infty\); if \(X\) is a reflexive space with the Kadec-Klee property, then every norm-one compact operator from \(X\) into a Banach space \(Y\) belongs to \(\mathcal{A}_{\|\cdot\|}(X,Y)\); a characterization of the diagonal operators belonging to \(\mathcal{A}_{\|\cdot\|}(X,X)\) when \(X=c_0\) or \(\ell_p\). Finally, in the case when \(X=Y\), a set analogous to \(\mathcal{A}_{\|\cdot\|}(X,X)\) but for the numerical radius instead of the operator norm is introduced and studied, showing some examples and interesting results on the relationship between these two sets.