The numerical radius and positivity of block matrices

TL;DR

This paper explores the positivity properties of block matrices and characterizes operators with a numerical radius bounded by one, providing both theoretical insights and an introductory overview.

Contribution

It offers a detailed exposition of block matrix positivity and introduces characterizations of operators with numerical radius constraints, bridging theory and foundational understanding.

Findings

Characterization of operators with numerical radius ≤ 1

Insights into positivity properties of block matrices

Foundational techniques for analyzing block matrix operators

Abstract

This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of block matrices, and especially their positivity properties. This is done by focussing on one major problem: characterisation of operators whose numerical radius is bounded by one. So, the article could serve as an introduction to that topic as well.