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On the geometry of $G$-norm

Lakshmi Kanta Dey, Subhadip Pal

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 \[ \|T\|_G := \inf_{δ>0}\sup\{\|Tx\|: \|x\|=1, \|Gx\|>1-δ\}, \] 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 $G$.

On the geometry of $G$-norm

Abstract

Let and be Banach spaces and let with . We study the geometry of -(semi-)norm on , defined by considering it as a norm (-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 coincides with the -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 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 -norm is achieved by the norm attainment set of a norm-attaining operator .
Paper Structure (5 sections, 17 theorems, 69 equations)

This paper contains 5 sections, 17 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Dual structure and smoothness of the $G$-norm
  5. Hilbert space applications and further properties

Key Result

Proposition 2.1

Let $G\in L(X,Y)$ with $\|G\|=1$. Then

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 3.1
  • ...and 25 more