On the geometry of $G$-norm

Abstract

Let $X$ and $Y$ be Banach spaces and let $G \in L(X,Y)$ with $\|G\|=1$. We study the geometry of $G$-(semi-)norm on $L(X,Y)$, defined by \[ \|T\|_G := \inf_{\delta>0}\sup\{\|Tx\|: \|x\|=1, \|Gx\|>1-\delta\}, \] considering it as a norm ($G$-norm), and further explore the associated numerical indices. In particular, we characterize relative spear operators, that is, operators for which the numerical radius with respect to $G$ coincides with the $G$-norm. Relations among the numerical indices and their invariance under isometric isomorphisms are established. We further obtain a description of the dual unit ball of $(L(X,Y),\|\cdot\|_G)$ and characterize smooth points of its unit ball. In finite-dimensional Hilbert spaces, we prove that relative spear operators are partial isometries. Finally, we establish some equivalent criteria for which the $G$-norm is achieved by the norm attainment set of a norm-attaining operator $G$.