Spectral families of projections in Hardy spaces

Abstract

Strongly continuous one-parameter groups of isometries in the reflexive Hardy spaces of the disc D and the half-plane are considered in the light of the author's previous joint result with H. Benzinger and T. A. Gillespie which generalizes Stone's theorem for unitary groups to arbitrary Banach spaces. It is shown that every such group { T t } of Hardy space isometries has a spectral decomposition (with respect to a suitable projection-valued function on the real line R ), as in the classical statement of Stone's theorem in Hilbert space. (The relevant type of projection-valued function is known as a “spectral family.”) This circle of ideas is intimately bound up with harmonic analysis, particularly in H p R . In particular, if the group { T t } acts in H p D and is associated as in Forelli's theorem with a group of parabolic Möbius transformations of D , then it can be analyzed by way of the translation group on H p R . The Stone-type spectral family of the latter is shown to be obtained by restriction of the M. Riesz projections to H p R . By this means a concrete description of the Stone-type spectral family for a parabolic isometric group on H p D is obtained. A pleasant by-product of the parabolic case is absorption of the classical Paley-Wiener theorem for H p R , 1 < p ⩽ 2, into the framework of the generalized Stone's theorem.