Inner functions and de Branges functions

TL;DR

This paper characterizes when an inner function in the upper half-plane can be expressed as E*/E for a de Branges function, establishing a correspondence between de Branges spaces and certain invariant subspaces.

Contribution

It provides a necessary and sufficient condition for inner functions to be represented via de Branges functions, linking de Branges spaces to invariant subspaces of H^2.

Findings

Characterization of inner functions as E*/E for de Branges functions.

Establishment of a bijective correspondence between de Branges spaces and subspaces S(F).

Connection of these subspaces to invariant subspaces in H^2(D).

Abstract

A necessary and sufficient condition for an inner function F in the upper half-plane (UHP) to satisfy F = E*/E where E is a de Branges function is presented. Since F_E =E^*/E is an inner function for any de Branges function E, and the map that takes f to f/E is an isometry of the de Branges space H(E) onto S(F_E), the orthogonal complement of F_E H^2, there is a natural bijective correspondence between de Branges spaces of entire functions and the set of subspaces S(F), for which F= E*/E for some de Branges function E. Under the canonical isometry of H^2(UHP) onto H^2(D) the subspaces S(F_E) become certain invariant subspaces for the backwards shift in H^2(D). I have been informed that the results contained in this paper are not new. Most of the results in this paper can be found, for example, in Theorem 2.7, Section 2.8, and Lemma 2.1 of V. Havin and J. Mashregi, "Admissable majorants for model spaces of H^2, Part I: slow winding of the generating inner function", Canad. J. Math. Vol. 55 (6), 2003 pp. 12311263. For this reason I have withdrawn this article.