De branges spaces of vector-valued functions

Abstract

This article is the first of a two-part series on de Branges spaces of vector valued functions and some of their applications to direct and inverse problems for canonical differential systems and Dirac-Kre?in systems. This first part is intended as an introduction to the spaces; the second, which is written as a continuation of the first, will focus on applications. The exposition is divided into a number of short sections, each of which deals with one topic. The list of topics covered in this part includes: reproducing kernel Hilbert spaces, J -inner matrix valued functions and some of their important subclasses, the Potapov-Ginzburg transform, linear fractional transformations, the de Branges spaces H(U), the de Branges space B(E).