Norm derivatives and geometry of bilinear operators

Abstract

We study the norm derivatives in the context of Birkhoff–James orthogonality in real Banach spaces. As an application of this, we obtain a complete characterization of the left-symmetric points and the right-symmetric points in a real Banach space in terms of the norm derivatives. We obtain a complete characterization of strong Birkhoff–James orthogonality in \(\ell _1^n\) and \(\ell _\infty ^n\) spaces. We also obtain a complete characterization of the orthogonality relation defined by the norm derivatives in terms of some newly introduced variation of Birkhoff–James orthogonality. We further study Birkhoff–James orthogonality, approximate Birkhoff–James orthogonality, smoothness and norm attainment of bounded bilinear operators between Banach spaces.