Dilation Theory, a one-century-old story

Abstract

This is a friendly recount of dilation theory in the complex plane. Starting with the pioneering result of F. Riesz (1923) in the Hardy space $H^p$, we provide a wide range of theorems which are uniformly spread over a century to tackle the problems in different function spaces such as the Bergman space, super-harmonically weighted Dirichlet spaces, the Bloch space, model spaces, de Branges-Rovnyak spaces, etc. We finish with some recent results and open questions, emerged in the last decade, to highlight the need for novel tools in dealing with such problems.