Dilation theory in finite dimensions: the possible, the impossible and the unknown

TL;DR

This paper explores the extent to which dilation theory can be developed within finite-dimensional linear algebra, revealing simple results, elementary proofs, and the limitations compared to infinite-dimensional operator theory.

Contribution

It provides new elementary proofs of inequalities and results in dilation theory using finite-dimensional linear algebra, clarifying the boundary between finite and infinite-dimensional phenomena.

Findings

Elementary proofs of von Neumann type inequalities

Identification of limits of finite-dimensional dilation techniques

Insights into differences between finite and infinite-dimensional operator theory

Abstract

This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These results can be used to give very elementary proofs of sharpened versions of some von Neumann type inequalities, as well as some other striking consequences about polynomials and matrices. Exploring the limits of the finite dimensional approach sheds light on the difference between those techniques and phenomena in operator theory that are inherently infinite dimensional, and those that are not.