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.

翻译：本文引入并研究环面上的一类算子类尺度，该尺度由Nagy与Foiaș提出的$\rho$-压缩算子$\mathcal{C}_{\rho}$类所启发。具体而言，我们的类是通过环面上双层位势积分算子的压缩性来定义的。我们证明，若进一步假设完全压缩性，则可获得涉及$\mathcal{C}_{\rho}$类特定变体的完整刻画。基于Crouzeix-Greenbaum与Schwenninger-de Vries的最新工作，我们还能得到相关的K-谱估计，从而推广环面上已有的文献结果。最后，我们展示一个特例，其中这些估计可被显著加强。