Operator representations of function algebras and functional calculus

Opuscula Mathematica

Adina Juratoni¹  Nicolae Suciu² 

[1] "Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei No. 2, Et. 2, 300006, Timişoara, Romania;West University of Timişoara, Department of Mathematics, Bv. V. Parvan 4, Timişoara 300223, Romania;

关键词: weak*-Dirichlet algebra;    Hardy space;    operator representation;    semispectral measure;   

DOI  :  http://dx.doi.org/10.7494/OpMath.2011.31.2.237

来源: DOAJ

【 摘 要 】

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 \(1\leq p\leq\infty\). Acharacterization 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 \(A\).

【 授权许可】

Unknown