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

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 Foiaş. 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 and improving existing results from the literature on the annulus. Finally, we exhibit a special case where these estimates can be significantly strengthened.