Abstract

A natural L^∞ functional calculus for an absolutely continuous contraction is investigated. It is harmonic in the sense that for such a contraction and any bounded measurable function φ on the circle, the image can rightly be considered as φ(T), where φ is the solution of the Dirichlet problem for the disk with boundary values φ. The main result shows that if the functional calculus is isometric on H^∞, then it is isometric on all of L^∞. As a consequence we obtain that if the contraction has an isometric H^∞ functional calculus and is in class C₀₀, then the range of the harmonic functional calculus is a hyperreflexive subspace of operators. In particular, the space of all Toeplitz operators with a bounded harmonic symbol acting on the Bergman space of the disc is hyperreflexive. Applications of these results to subnormal operators are also presented.

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