Overview
Syllabus
Intro
Classical motivation
Goals of the talk
Representation of Pick functions
Connection with positive matrices
Matrix inequalities
The functional calculus
Special matrix functions
Classical representations: Löwner-Nevanlinna
Classical representations: Kraus
Thematic ideas in classical proofs
The matrix universe (more usual)
Noncommutative sets
Noncommutative functions
nc matrix functions
Löwner's Theorem
free Nevanlinna representation
What is the royal road?
Royal road approach
A royal road: dominions
Royal road theorem
Proof idea
A road to realizations
Convex functions
Kraus theorem
The butterfly realization
Further considerations
Taught by
Fields Institute