Dilation Theory Yesterday and Today

Paul Halmos’ work in dilation theory began with a question and its answer: Which operators on a Hilbert space H can be extended to normal operators on a larger Hilbert space K ⊇ H? 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.