The Bishop-Phelps-Bollobás property for numerical radius in \(\ell _{1}(\mathbb {C})\) (Q2856670)

For a Banach space \(X\), denote by \(L(X)\) the space of all bounded linear operators \(T: X \to X\), and NEWLINE\[NEWLINE\Pi(X)=\left\{(x,x^*)\in X\times X^*: \|x\|=\|x^*\|=x^*(x)=1\right\}.NEWLINE\]NEWLINE The numerical radius of \(T \in L(X)\) is the following quantity: NEWLINE\[NEWLINE\nu(T) = \sup\left\{|x^*(Tx)|: (x,x^*)\in \Pi(X)\right\}.NEWLINE\]NEWLINENEWLINENEWLINEMotivated by the Bishop-Phelps-Bollobás theorem (see [\textit{B. Bollobás}, Bull. Lond. Math. Soc. 2, 181--182 (1970; Zbl 0217.45104)]), the authors introduce the following concept: \(X\) is said to have the Bishop-Phelps-Bollobás property for the numerical radius (BPBp-\(\nu\) for short) if, for every \(\varepsilon \in (0, 1)\), there exists \(\delta > 0\) such that, for any given \(T \in L(X)\) with \(\nu(T) = 1\) and a pair \((x,x^*)\in \Pi(X)\) satisfying \(|x^*(Tx)| > 1 - \delta\), there exist \(S \in L(X)\) with \(\nu(S) = 1\) and a pair \((y,y^*)\in \Pi(X)\) such that \(\nu(T - S) \leq \varepsilon\), \(\|x - y \| \leq \varepsilon\), \(\|x^* - y^* \| \leq \varepsilon\) and \(|y^*(Sy)| = 1\). It is shown that the spaces \(\ell_1\) and \(c_0\) possess the BPBp-\(\nu\).