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A characterization of Banach spaces with numerical index one

Subhadip Pal, Saikat Roy, Debmalya Sain

Abstract

We investigate the extremal properties of the unit ball of $L(X)_w^*$, the dual space of bounded linear operators defined on a Banach space $X$ equipped with the numerical radius norm. As an application of the present study, we obtain a geometric characterization of Banach spaces with numerical index one, which extends the well-known McGregor's characterization of finite-dimensional Banach spaces with numerical index one. We also present refinements of several earlier results in this direction, including an explicit description of the extreme points of $B_{L(X)_w^*}$, the unit ball of $L(X)_w^*$, for any finite-dimensional Banach space $X$. This allows us to obtain an independent and elementary proof of McGregor's characterization of finite-dimensional Banach spaces with numerical index one.

A characterization of Banach spaces with numerical index one

Abstract

We investigate the extremal properties of the unit ball of , the dual space of bounded linear operators defined on a Banach space equipped with the numerical radius norm. As an application of the present study, we obtain a geometric characterization of Banach spaces with numerical index one, which extends the well-known McGregor's characterization of finite-dimensional Banach spaces with numerical index one. We also present refinements of several earlier results in this direction, including an explicit description of the extreme points of , the unit ball of , for any finite-dimensional Banach space . This allows us to obtain an independent and elementary proof of McGregor's characterization of finite-dimensional Banach spaces with numerical index one.

Paper Structure

This paper contains 2 sections, 18 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Extreme points of the unit ball in the dual of certain operator spaces under the numerical radius norm

Key Result

Theorem 1.1

MG Let $X$ be a finite-dimensional Banach space. Then $n(X)=1$ if and only if $|x^*(x)|=1$ for every $(x, x^*)\in E_{B_X}\times E_{B_{X^*}}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Remark 2.5
  • ...and 19 more