Numerical ranges of model operators and Clark measures

Abstract

Let ϑ and η be inner functions with two spectral points on the unit circle T. Assuming ϑ(0)=η(0)=0, let ϑ0=ϑ/χ and η0=η/χ. Generalizing former results in [15] we prove that coincidence of some tangent polygons to the numerical ranges W(S(ϑ0)) and W(S(η0)) are already sufficient for coincidence of the model operators S(ϑ0) and S(η0). Instead of exponential representations of analytic self-maps of the upper half-plane, we rely here on Cauchy transforms of Clark measures associated with ϑ and η. Relaxing the assumptions yielding uniqueness, conditions are given also for the existence of multipliers between the model spaces H(ϑ) and H(η).