De Branges spaces and Krein's theory of entire operators

TL;DR

This paper explores the deep connection between Krein's entire operators and de Branges' Hilbert spaces of entire functions, highlighting their interplay and applications in spectral theory through a modern functional model.

Contribution

It provides a contemporary synthesis of Krein's and de Branges' theories, illustrating their interrelation and recent developments in spectral analysis.

Findings

Establishes a functional model linking the two theories

Demonstrates applications to spectral theory of difference operators

Provides new insights into the structure of entire operators

Abstract

This work 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 work 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.