TY  - JOUR
ID  - Juratoni2011
LB  - Juratoni2011
AU  - Juratoni, Adina
AU  - Suciu, Nicolae
TI  - Operator representations of function algebras and functional calculus
ST  - Operator representations of function algebras and functional calculus
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2011
VL  - 31
IS  - 2
SP  - 237
EP  - 255
DO  - http://dx.doi.org/10.7494/OpMath.2011.31.2.237
UR  - http://dx.doi.org/10.7494/OpMath.2011.31.2.237
AB  - 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 \(A\).
KW  - weak*-Dirichlet algebra
KW  - Hardy space
KW  - operator representation
KW  - semispectral measure
SN  - 1232-9274
ER  -