@Article{Juratoni2011,
author = "Juratoni, Adina and Suciu, Nicolae",
title = "Operator representations of function algebras and functional calculus",
journal = "Opuscula Math.",
year = "2011",
volume = "31",
number = "2",
pages = "237--255",
doi = "http://dx.doi.org/10.7494/OpMath.2011.31.2.237",
url = "http://dx.doi.org/10.7494/OpMath.2011.31.2.237",
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 \(C_\rho\) 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 \(A\).",
keywords = "weak*-Dirichlet algebra, Hardy space, operator representation, semispectral measure",
issn = "1232-9274"
}