On the geometry of $G$-norm

TL;DR

This paper investigates the geometric properties of the G-norm on operator spaces between Banach spaces, characterizing special operators, their numerical indices, dual structures, and conditions for norm attainment.

Contribution

It introduces the concept of G-norms on operator spaces, characterizes relative spear operators, and explores their geometric and dual properties, including in finite-dimensional Hilbert spaces.

Findings

Characterization of relative spear operators as partial isometries in Hilbert spaces.

Relations between numerical indices and invariance under isometric isomorphisms.

Description of the dual unit ball and smooth points in the G-normed space.

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 $G$.