Dilation theory yesterday and today

TL;DR

This paper reviews the historical development and current state of dilation theory in operator theory, highlighting its significance in representing complex operators via simpler, larger Hilbert space structures.

Contribution

It provides a comprehensive summary of the evolution, key results, and ongoing developments in dilation theory, emphasizing its central role in noncommutative analysis.

Findings

Dilation theory connects operators to normal operators on larger spaces.

The field has evolved from foundational questions to diverse modern results.

Dilation concepts are fundamental in noncommutative analysis.

Abstract

Paul Halmos' work in dilation theory began with a question and its answer: Which operators on a Hilbert space can be extended to normal operators on a larger Hilbert space? The answer is interesting and subtle. The idea of representing operator-theoretic structures in terms of conceptually simpler structures acting on larger Hilbert spaces has become a central one in the development of operator theory and, more generally, noncommutative analysis. The work continues today. In this article we summarize some of these diverse results and their history.