Operator representations of function algebras and functional calculus - Adina Juratoni; Nicolae Suciu (Opuscula Mathematica, 2011)
This paper deals with some operator representations \(Φ\) 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 \(1≤ p≤∞\). A characterization for the existence of an extension \(Φ_p\) of \(Φ\) to \(L^p(m)\) is given in the terms of a semispectral measure \(F_Φ\) of \(Φ\). 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 \(Φ_p|H^p(m)\) 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 \(F_Φ\) 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 \(A\).