Operator representations of function algebras and functional calculus

Abstract

This paper deals with some operator representations $\Phi$ of a weak*-Dirichlet algebra $A$, which can be extended to the Hardy spaces $H^{p}(m)$, associated to $A$ and to a representing measure m of $A$, for $1\leq p\leq\infty$. A characterization for the existence of an extension $\Phi_p$ of $\Phi$ to $L^{p}(m)$ is given in the terms of a semispectral measure $F_\Phi$ of $\Phi$. For the case when the closure in $L^{p}(m)$ of the kernel in $A$ of $m$ is a simply invariant subspace, it is proved that the map $\Phi_p|H^p(m)$ can be reduced to a functional calculus, which is induced by an operator of class $C_ρ$ in the Nagy-Foiaş sense. A description of the Radon-Nikodym derivative of $F_\Phi$ 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 $L^{p}(m)$ norm, form the range of an embedding of the open unit disc into a Gleason part of $A$.