  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.
