The Harmonic Functional Calculus and Hyperreflexivity

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 φ̂(T ), where φ̂ 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.