Operator theory induced by powers of the de Branges-Rovnyak kernel and its application

TL;DR

This paper investigates properties of de Branges-Rovnyak kernels, demonstrating that their exponential is strictly positive definite when associated Schur functions have nontrivial inner parts, with implications for operator theory.

Contribution

It introduces a new property of de Branges-Rovnyak kernels and establishes conditions for their exponential to be strictly positive definite based on Schur class functions.

Findings

Exponential of de Branges-Rovnyak kernel is strictly positive definite under certain conditions.

Provides new insights into operator theory related to de Branges-Rovnyak kernels.

Connects kernel properties with the inner part of Schur class functions.

Abstract

In this note, we give a new property of de Branges-Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges-Rovnyak kernel is strictly positive definite if the inner part of the corresponding Schur class function is nontrivial.