Szeg\H{o}'s Theorem for Canonical Systems: the Arov Gauge and a Sum Rule

TL;DR

This paper explores Szeg\

Contribution

It introduces a new sum rule for canonical systems in the Arov gauge, linking entropy integrals to system coefficients, advancing spectral theory understanding.

Findings

Entropy integral equals an integral over system coefficients in Arov gauge

Established a sum rule connecting spectral data and system parameters

Enhanced the spectral theory framework for canonical systems

Abstract

We consider canonical systems and investigate the Szeg\H{o} class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the canonical system. This sum rule provides a spectral theory gem in the sense proposed by Barry Simon.