Opuscula Mathematica
Opuscula Math. 31, no. 2 (), 237-255
Opuscula Mathematica

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 \(C_\rho\) 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.
Mathematics Subject Classification: 46J25, 47A20, 46J10.
Cite this article as:
Adina Juratoni, Nicolae Suciu, Operator representations of function algebras and functional calculus, Opuscula Math. 31, no. 2 (2011), 237-255, http://dx.doi.org/10.7494/OpMath.2011.31.2.237
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ISSN 1232−9274, e-ISSN 2300−6919, DOI http://dx.doi.org/10.7494/OpMath
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