Dilation theory A century of distancing from the frontiers

Abstract

Analytic functions, despite exhibiting very satisfactory behavior inside their domains of definition, e.g., being infinitely differentiable and having power series representations, could be quite wild and show bizarre performance at the boundary points. Not being continuous at a single boundary point is the simplest of such phenomena. However, much more pathological behaviors are foreseeable in the frontiers. That is why boundary behavior studies have faced several obstacles since early days of complex analysis, even though in many cases the final outcome is very appealing and usually ends with profound and beautiful mathematical results. Dilation is an old technique that mathematicians have exploited to deviate from the frontiers and go deep inside the domain. Then, standing within the domain and coming back from inside toward the frontiers, since our footsteps are in a solid territory on which the function behaves well and is fully under the control, one tries to better understand the boundary behavior at the frontiers. No wonder the story has many ups and downs and is filled with numerous successes and failures. Starting with a pioneering theorem of F. Riesz in 1923, we provide a wide range of results which are uniformly spread over a century to tell parts of the dilation history in the complex plane. We finish with some recent results and open questions, obtained in the course of the last decade, to highlight the need for novel tools in studying newly emerged function spaces and the boundary behavior of their elements.