A Journey into Matrix Analysis

TL;DR

This thesis explores advanced topics in matrix analysis, inequalities, decompositions, and operator theory, providing new insights, conjectures, and methods relevant for researchers and students in pure and applied mathematics.

Contribution

It introduces novel techniques in matrix inequalities, functional calculus, and matrix decompositions, and offers a new proof of the Spectral Theorem derived from matrix cases.

Findings

Development of new matrix inequality techniques

Introduction of a unitary orbit method for functional calculus

A novel proof of the Spectral Theorem from matrix analysis

Abstract

This is the Habilitation Thesis manuscript presented at Besan\c{c}on on January 5, focusing on Matrix Analysis, Matrix Inequalities and Matrix Decompositions. There are also some topics in (Hilbert space) Operator Theory. The text should be of interest for a large audience of researchers and students in pure and applied mathematics. We may divide it into five parts: 1) Chapter 1 is an introductory chapter, some results from the period 1999-2010 are given, and a few conjectures are proposed. 2) Chapters 2-4 deal with matrix inequalities, Chapter 2 is concerned with norm inequalities and logmajorization and Chapters 3-4 with functional calculus and a unitary orbit technique that I started to develop in 2003. 3) Chapter 5 is a time-break in infinite dimensional Hilbert space operators, the essential numerical range plays a key role. 4) Chapters 6-8 establish several decompositions for partitioned matrices, especially for positive block matrices. Some norm inequalities involving the numerical range are derived. 5) Chapter 9 may be of special interest for students : a proof of the Spectral Theorem for bounded operators is derived from the matrix case.