Cauchy-de Branges spaces, geometry of their reproducing kernels and multiplication operators

TL;DR

This paper explores the geometric structure and multiplication operator properties of Cauchy-de Branges spaces, extending classical de Branges theory to a broader setting involving entire functions and discrete measures.

Contribution

It introduces geometric and operator-theoretic extensions of de Branges space properties to Cauchy-de Branges spaces, including bases of reproducing kernels and domain density characterizations.

Findings

Analysis of completeness and Riesz bases of reproducing kernels.

Characterization of the density of the multiplication domain.

Extension of classical de Branges properties to a more general setting.

Abstract

Cauchy-de Branges spaces are Hilbert spaces of entire functions defined in terms of Cauchy transforms of discrete measures on the plane and generalizing the classical de Branges theory. We consider extensions of two important properties of de Branges spaces to this, more general, setting. First, we discuss geometric properties (completeness, Riesz bases) of systems of reproducing kernels corresponding to the zeros of certain entire functions associated to the space. In the case of de Branges spaces they correspond to orthogonal bases of reproducing kernels. The second theme of the paper is a characterization of the density of the domain of multiplication by $z$ in Cauchy-de Branges spaces.