Opuscula Math. 31, no. 2 (2011), 237-255

Opuscula Mathematica

Operator representations of function algebras and functional calculus

Adina Juratoni
Nicolae Suciu

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.

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  • Adina Juratoni
  • "Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei No. 2, Et. 2, 300006, Timişoara, Romania
  • Nicolae Suciu
  • West University of Timişoara, Department of Mathematics, Bv. V. Parvan 4, Timişoara 300223, Romania
  • Received: 2010-05-18.
  • Revised: 2010-07-13.
  • Accepted: 2010-07-15.
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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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