On numerical radius and Crawford number attainment sets of a bounded linear operator

TL;DR

This paper provides a complete characterization of the attainment sets for numerical radius and Crawford number of bounded linear operators on Hilbert spaces, exploring their properties and differences from general normed spaces.

Contribution

It offers a comprehensive description of the attainment sets for numerical radius and Crawford number, and compares their properties in Hilbert and normed spaces.

Findings

Complete characterization of Crawford number attainment set

Analysis of intersection properties of attainment sets

Differences between Hilbert and normed space operators

Abstract

We completely characterize the Crawford number attainment set and the numerical radius attainment set of a bounded linear operator on a Hilbert space. We study the intersection properties of the corresponding attainment sets of numerical radius, Crawford number, norm, minimum norm of a bounded linear operator defined on a normed space. Our study illustrates the similarities and the differences of the extremal properties of a bounded linear operator on a Hilbert space and a general normed space.