We introduce and study a scale of operator classes on the annulus that is motivated by the Cρ classes of ρ-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 Cρ 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.
Let Ar = {r < |z| < 1} be an annulus. We consider the class of operatorsand show that for every bounded holomorphic function φ on Ar : supwhere the constant √ 2 is the best possible. We do this by characterizing the calcular norm induced on H ∞ (Ar) by Fr as the multiplier norm of a suitable holomorphic function space on Ar.
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