Positivity conditions on the annulus via the double-layer potential kernel

TL;DR

This paper introduces new operator classes on the annulus based on the double-layer potential kernel, providing characterizations, spectral estimates, and strengthening results under certain conditions.

Contribution

It defines a scale of operator classes related to the double-layer potential and characterizes them using $ ho$-contraction concepts, extending spectral estimate results.

Findings

Characterization of operator classes via double-layer potential

Extension of spectral estimates to the annulus setting

Strengthened estimates in specific cases

Abstract

We introduce and study a scale of operator classes on the annulus that is motivated by the $\mathcal{C}_{\rho}$ classes of $\rho$-contractions of Nagy and Foia\c{s}. In particular, our classes are defined in terms of the contractivity of the double-layer potential integral operator over the annulus. We prove that if, in addition, complete contractivity is assumed, then one obtains a complete characterization involving certain variants of the $\mathcal{C}_{\rho}$ classes. Recent work of Crouzeix-Greenbaum and Schwenninger-de Vries allows us to also obtain relevant K-spectral estimates, generalizing existing results from the literature on the annulus. Finally, we exhibit a special case where these estimates can be significantly strengthened.