By definition, a pair \((X,Y)\) of Banach spaces has the Bishop-Phelps-Bollobás property (BPBp for short) if for every \(\varepsilon>0\) there exists \(\eta(\varepsilon)>0\) such that for every operator \(T\in \mathcal{L}(X,Y)\) with \(\|T\|=1\) and \(x_0\in S_X\) such that \(\|Tx_0\|>1-\eta(\varepsilon)\), there exist \(S\in \mathcal{L}(X,Y)\) and \(x\in S_X\) with \[ \|S\|=\|Sx\|=1, \quad \|S-T\|<\varepsilon, \quad \text{and} \quad \|x_0-x\|<\varepsilon. \tag{*} \] Among many other results, it is known that the BPBp holds true when \(Y\) has Lindenstrauss’s property \(\beta\) [M. D. Acosta et al., J. Funct. Anal. 254, No. 11, 2780–2799 (2008; Zbl 1152.46006)], in particular, if \(c_0\subset Y \subset \ell_\infty\) or if \(Y\) is a finite-dimensional polyhedral space. On the other hand, there are Banach spaces \(Y\) such that the pair \((\ell_1^2,Y)\) fails the BPBp even though all elements in \(\mathcal{L}(\ell_1^2,Y)\) attain their norms [loc.cit.]; moreover, this \(Y\) can be found in such a way that for every Banach space \(X\) norm attaining operators are dense in \(\mathcal{L}(X,Y)\) [R. Aron et al., Trans. Am. Math. Soc. 367, No. 9, 6085–6101 (2015; Zbl 1331.46008)].
When a pair \((X,Y)\) does not enjoy the BPBp, one can ask what additional properties of \(T\in \mathcal{L}(X,Y)\) may ensure the existence of approximation (\(*\)). An important example of such kind of theorems was given by R. M. Aron et al. [Proc. Am. Math. Soc. 139, No. 10, 3553–3560 (2011; Zbl 1235.46013)], where it was demonstrated that for arbitrary \(X\) and for \(Y = C(K)\) the approximation exists for Asplund operators. The paper under review is developing the latter line of research. The authors introduce the notion of generic Fréchet differentiability operator. This name is given to those \(T\in \mathcal{L}(X,Y)\) for which the map \(x \mapsto \|T(x)\|\) is Fréchet differentiable on a dense \(G_\delta\) subset of \(X\). They study properties of this class, the relationship with Asplund operators and with \(\Gamma\)-flat operators, introduced in [B. Cascales et al., J. Funct. Anal. 274, No. 3, 863–888 (2018; Zbl 1396.46006)]. This concept allows to give a new proof of the Aron-Cascales-Kozhushkina theorem [Aron et al., loc.cit.]with the sharp estimate \(\eta(\varepsilon) = \varepsilon^2/2\). The authors also demonstrate a dual version of the Bishop-Phelps-Bollobás property for strong Radon-Nikodým operators \(T: \ell_1 \to Y\).
The paper contains all the necessary preliminaries for its understanding.