De Branges Spaces and Krein’s Theory of Entire Operators

Abstract

This chapter presents a contemporary treatment of Krein’s entire operators with deficiency indices (1, 1) and de Branges’ Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected, and the interplay between them has had an impact in both spectral theory and the theory of functions. This chapter exhibits the interrelation between Krein’s and de Branges’ theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.