Joint numerical radius of Tuples: Extreme points, subdifferential set and Gateaux derivative

TL;DR

This paper investigates the geometric and differential properties of the joint numerical radius in finite-dimensional operator tuples, providing characterizations of extreme points, subdifferentials, smoothness, orthogonality, and derivatives.

Contribution

It introduces a detailed structure of extreme points of the dual unit ball and derives the Gateaux derivative of the joint numerical radius for operator tuples.

Findings

Characterization of extreme points of the dual unit ball.

Explicit expression for the subdifferential set of the joint numerical radius.

Calculation of the Gateaux derivative of the joint numerical radius.

Abstract

Suppose $\mathcal{Z}$ is the space of all tuples of operators on a finite-dimensional Banach space endowed with the joint numerical radius norm. We obtain the structure of the extreme points of the dual unit ball of $\mathcal{Z}.$ Using this, we derive an expression for the subdifferential set of the joint numerical radius of a tuple in $\mathcal{Z}.$ Applying this expression, we characterize smooth tuples and Birkhoff-James orthogonality in $\mathcal{Z}.$ Finally, we obtain the Gateaux derivative of the joint numerical radius of a tuple.