Factorization in Krein Spaces

Abstract

The factorization theory of analytic functions is an aspect of complex analysis which owes its modern formulation to the work of Beurling on invariant subspaces of the shift. Certain Hilbert spaces of analytic functions which result from his ideas were constructed by de Branges and Rovnyak. The resulting interplay between complex analysis and operator theory remains a dominant theme of contemporary analysis. The proof of the Bieberbach conjecture and an attack on the Riemann hypothesis are some of the more striking examples of these concepts. These results are obtained in the background of a remarkable interpolation theory for analytic functions, not only with complex values, but also with values in a Krein space. The concept of a unitary linear system allows a characterization of the basic Hilbert spaces of analytic functions of the interpolation theory.