Stein's theorem on the interpolation of a family of operators between two analytic spaces is generalized, both to a multiply connected domain and to an interpolation between more than two spaces. The theorem is then applied to get setwise upper bounds for spectra of convolution operators on Lp of the circle. In particular the spectra of operators given by convolution by Cantor-Lebesgue-type measures on Lp are determined. The same is done for certain Riesz products. These results are used to derive a result on translation-invariant subspaces of Lp of the circle.
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