Operator representations of function algebras and functional calculus
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...
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doaj-af7f3d53e8814922a8e1dad363281d5a2020-11-24T23:01:25ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742011-01-01312237255http://dx.doi.org/10.7494/OpMath.2011.31.2.2373116Operator representations of function algebras and functional calculusAdina Juratoni0Nicolae Suciu1"Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei No. 2, Et. 2, 300006, Timişoara, RomaniaWest University of Timişoara, Department of Mathematics, Bv. V. Parvan 4, Timişoara 300223, RomaniaThis 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 \(A\).http://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3116.pdfweak*-Dirichlet algebraHardy spaceoperator representationsemispectral measure |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Adina Juratoni Nicolae Suciu |
spellingShingle |
Adina Juratoni Nicolae Suciu Operator representations of function algebras and functional calculus Opuscula Mathematica weak*-Dirichlet algebra Hardy space operator representation semispectral measure |
author_facet |
Adina Juratoni Nicolae Suciu |
author_sort |
Adina Juratoni |
title |
Operator representations of function algebras and functional calculus |
title_short |
Operator representations of function algebras and functional calculus |
title_full |
Operator representations of function algebras and functional calculus |
title_fullStr |
Operator representations of function algebras and functional calculus |
title_full_unstemmed |
Operator representations of function algebras and functional calculus |
title_sort |
operator representations of function algebras and functional calculus |
publisher |
AGH Univeristy of Science and Technology Press |
series |
Opuscula Mathematica |
issn |
1232-9274 |
publishDate |
2011-01-01 |
description |
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 \(A\). |
topic |
weak*-Dirichlet algebra Hardy space operator representation semispectral measure |
url |
http://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3116.pdf |
work_keys_str_mv |
AT adinajuratoni operatorrepresentationsoffunctionalgebrasandfunctionalcalculus AT nicolaesuciu operatorrepresentationsoffunctionalgebrasandfunctionalcalculus |
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