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.
