An invitation to dilation theory

The basic idea behind dilation theory is to represent a fairly general operator \(T\) on a Hilbert space as a piece of a better understood operator \(N\) on a larger Hilbert space; the operator \(N\) is then called a dilation of \(T\). The prototypical example is Sz.-Nagy’s dilation theorem, which says that every operator of norm at most one admits a unitary dilation. This talk gives an introduction to the subject and explains how it connects to operator algebra theory.

Further reading: Paulsen, Completely Bounded Maps and Operator Algebras; Shalit, “Dilation theory: a guided tour” (arXiv:2002.05596); Fackler–Glück, “A toolkit for constructing dilations on Banach spaces” (arXiv:1709.08547).