Operator Representations of Function Algebras and Functional Calculus

This paper deals with some operator representations Φ of a weak*-Dirichlet algebra A, which can be extended to the Hardy spaces H(m), associated to A and to a representing measure m of A, for 1 ≤ p ≤ ∞. A characterization for the existence of an extension Φp of Φ to L(m) is given in the terms of a semispectral measure FΦ of Φ. For the case when the closure in L(m) of the kernel in A of m is a simply invariant subspace, it is proved that the map Φp|H(m) can be reduced to a functional calculus, which is induced by an operator of class Cρ in the Nagy-Foiaş sense. A description of the Radon-Nikodym derivative of FΦ is obtained, and the log-integrability of this derivative is proved. An application to the scalar case, shows that the homomorphisms of A which are bounded in L(m) norm, form the range of an embedding of the open unit disc into a Gleason part of A.