On the Bishop-Phelps-Bollobás theorem for operators and numerical radius (Q2809359)

This paper is devoted to study the Bishop-Phelps-Bollobás property for the numerical radius (BPBp-nu for short) and its relationship with the actively studied Bishop-Phelps-Bollobás property for operators. The BPBp-nu, which was recently introduced in [\textit{A. J. Guirao} and \textit{O. Kozhushkina}, Stud. Math. 218, No. 1, 41--54 (2013; Zbl 1285.47008)], deals with the denseness of the numerical radius attaining operators between two Banach spaces \(X\) and \(Y\) from a quantitative point of view. The authors give sufficient conditions on \(X\) and \(Y\), which involve the BPBp-nu, in order to obtain that the pair of \((X,Y)\) has the Bishop-Phelps-Bollobás property for operators. More precisely, it is proved that, whenever \(X\) is strongly lush and \(X\oplus_1 Y\) has the BPBp-nu or \(Y\) is strongly lush and \(X\oplus_{\infty}Y\) has the BPBp-nu, then the pair \((X,Y)\) has the Bishop-Phelps-Bollobás property for operators (in fact, instead of the BPBp-nu, the authors use a weakening of the BPBp-nu which avoids a normalization). Strong lushness is a property of geometric nature shared by \(L_1(\mu)\) spaces, \(C(K)\) spaces, and finite-codimensional subspaces of \(C(K)\) spaces for instance. Therefore, the mentioned results generalize the fact that \((L_1(\mu),Y)\) has the Bishop-Phelps-Bollobás property for operators whenever \(L_1(\mu)\oplus_1 Y\) has the BPBp-nu which was shown in [the authors, ``On the Bishop-Phelps-Bollobás property for numerical radius'', Abstr. Appl. Anal. 2014, Article ID 479208, 15 p. (2014; \url{doi:10.1155/2014/479208})].