Opuscula Math. 31, no. 2 (2011), 237-255
http://dx.doi.org/10.7494/OpMath.2011.31.2.237

Opuscula Mathematica

Operator representations of function algebras and functional calculus

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.

Full text (pdf)