Extreme points of the unit ball of $\mathcal{L}(X)_w^*$ and best approximation in $\mathcal{L}(X)_w$

TL;DR

This paper investigates the geometric structure of the space of bounded linear operators on a Banach space under the numerical radius norm, characterizing extreme points, orthogonality, and approximation properties.

Contribution

It provides a detailed description of the extreme points of the dual unit ball and links orthogonality and approximation in operator and Banach space contexts.

Findings

Characterization of extreme points of the dual unit ball.

Connection between Birkhoff-James orthogonality in operators and in the Banach space.

Distance formulas and approximation results for operators with smoothness properties.

Abstract

We study the geometry of $\mathcal{L}(X)_w,$ the space of all bounded linear operators on a Banach space $X,$ endowed with the numerical radius norm, whenever the numerical radius defines a norm. We obtain the form of the extreme points of the unit ball of the dual space of $\mathcal{L}(X)_w.$ Using this structure, we explore Birkhoff-James orthogonality, best approximation and deduce distance formula in $\mathcal{L}(X)_w.$ A special attention is given to the case of operators satisfying a notion of smoothness. Finally, we obtain an equivalence between Birkhoff-James orthogonality in $\mathcal{L}(X)_w$ and that in $X.$