On joint numerical radius of operators and joint numerical index of a Banach space

TL;DR

This paper introduces the concepts of joint numerical range, joint numerical radius, and joint numerical index for operators on Banach spaces, analyzing their properties and computing these indices for classical spaces.

Contribution

It generalizes numerical range and radius to tuples of operators, studies their properties, and introduces the joint numerical index of Banach spaces with explicit calculations.

Findings

Joint numerical radius defines a norm iff the numerical radius does.

On finite-dimensional spaces, joint numerical radius can be characterized by extreme points.

The joint numerical index is computed for several classical Banach spaces.

Abstract

Generalizing the notion of numerical range and numerical radius of an operator on a Banach space, we introduce the notion of joint numerical range and joint numerical radius of tuple of operators on a Banach space. We study the convexity of the joint numerical range. We show that the joint numerical radius defines a norm if and only if the numerical radius defines a norm on the corresponding space. Then we prove that on a finite-dimensional Banach space, the joint numerical radius can be retrieved from the extreme points. Furthermore, we introduce a notion of joint numerical index of a Banach space. We explore the same for direct sum of Banach spaces. Applying these results, finally we compute the joint numerical index of some classical Banach spaces.