Weak-star quasi norm attaining operators (Q6084834)

Let \(T \in \mathcal{L}(X,Y)\) be a bounded linear operator between the Banach spaces \(X\) and \(Y\) over \(\mathbb{K} = \mathbb{R}\) or \(\mathbb{C}\). We say that \(T\) quasi attains its norm if there is a sequence \((y_n) \subseteq T(B_X)\) such that \(\lim_n y_n \in \|T\|S_Y\). In this case, we write \(T \in \mathrm{QNA}(X,Y)\). After a systematic study of such a property in [\textit{G.~Choi} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116, No.~3, Paper No.~133, 32~p. (2022; Zbl 1500.46010)], the authors of the present paper are interested in introducing and studying a similar property, where now a coarser topology in the range space \(Y\) is considered. More precisely, the weak-star topology is considered in \(Y\) when \(Y\) is a dual space. We then say that \(T \in \mathcal{L}(X, Y^*)\) weak-star attains its norm, i.e., \(T \in w^* \mathrm{QNA}(X, Y^*)\), if there are a net \((x_{\alpha}) \subseteq B_X\) and a vector \(u^* \in \|T\| S_{Y^*}\) such that \(w^*\)-\(\lim_{\alpha} x_{\alpha} = u^*\). In other words, this means that \(\overline{T(B_X)}^{w^*} \cap \|T\|S_{Y^*} \not= \emptyset\). It turns out that the set \(w^* \mathrm{QNA}(X, Y^*)\) is (norm) dense in \(\mathcal{L}(X, Y^*)\) for all Banach spaces \(X\) and \(Y\) (see Theorem~2.1). This goes completely in the opposite direction of the classical (quasi) norm attainment (see also Example~2.5). The authors also prove that the quasi and the weak-star quasi norm attainment are equivalent properties for weakly compact operators (see Corollary~3.2). There is also an interesting example (see Remark~3.5) that shows that a weak-star quasi norm-attaining monomorphism is not necessarily norm-attaining. They conclude the paper by studying when an equality holds true in the following chain \[ \mathrm{NA}(X, Y^*) \subseteq \mathrm{QNA}(X, Y^*) \subseteq w^* \mathrm{QNA}(X,Y^*) \subseteq \mathcal{L}(X, Y^*). \]