Krein-type theorems and ordered structure for Cauchy-de Branges spaces

TL;DR

This paper extends Krein's theorems to certain entire functions represented as ratios of discrete Cauchy transforms, leading to new ordering results for subspaces in Hilbert spaces of entire functions.

Contribution

It introduces novel extensions of Krein's theorems and de Branges' Ordering Theorem for a broader class of entire functions and subspaces.

Findings

New versions of de Branges' Ordering Theorem are established.

Examples demonstrate the sharpness of the new results.

Extensions apply to ratios of discrete Cauchy transforms.

Abstract

We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.