On a conjecture of de Branges

TL;DR

This paper presents a broad class of counterexamples that challenge a conjecture by de Branges, specifically questioning the necessity of the continuity property in defining de Branges spaces.

Contribution

It introduces new counterexamples that disprove a longstanding conjecture about the axiomatic foundations of de Branges spaces.

Findings

Counterexamples show the continuity property is not necessary in de Branges spaces

Challenges the assumption that the conjecture holds universally

Provides insights into the structure of de Branges spaces

Abstract

We provide a broad class of counterexamples to a conjecture of L. de Branges concerning the superfluity of the continuity property in the axiomatic description of de Branges spaces.