This paper deals with some operator representations φ of a weak*-Dirichlet algebra A, which can be extended to the Hardy spaces Hp(m), associated to A and to a representing measure m of A, for 1 ≤ p ≤ ∞. A characterization for the existence of an extension φp of φ to Lp(m) is given in the terms of a semispectral measure Fφ of φ. For the case when the closure in Lp(m) of the kernel in A of m is a simply invariant subspace, it is proved that the map φp/Hp(m) can be reduced to a functional calculus, which is induced by an operator of class Cρ in the Nagy-Foias sense. A description of the Radon-Nikodym derivative of Fφ 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 Lp(m) norm, form the range of an embedding of the open unit disc into a Gleason part of A.
In this paper the canonical representation of an A-contraction T on a Hilbert space H is used to obtain some conditions concerning the concept of A-ergodicity studied in [14-17]. The regular case and the case of R(A) closed are considered, and specifically, the TT*-contractions are studied. Some spectral properties are also given for certain particular class of A-isometries.
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