An Alternate Proof of De Branges Theorem on Canonical Systems

TL;DR

This paper offers an alternative proof of De Branges' theorem by analyzing the spectral properties of canonical systems in the limit circle case, establishing the constancy of the defect index.

Contribution

It introduces a new approach to proving De Branges' theorem using spectral theory of linear relations in canonical systems.

Findings

The defect index is constant on C in the limit circle case.

Canonical systems with trace H(x)=1 imply the limit point case.

Provides an alternative proof of De Branges' theorem.

Abstract

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on C. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H(x)=1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.