Spectral families of projections in Hardy spaces (Q759984)

Strongly continuous one-parameter groups of isometries in the reflexive Hardy spaces of the disc \({\mathbb{D}}\) and the half-plane are considered in the light of the author's previous joint result with \textit{H. Benzinger} and \textit{T. A. Gillespie} [Trans. Am. Math. Soc. 275, 431-475 (1983; Zbl 0509.47028)] 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 \({\mathbb{R}})\), 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 \(H^ p({\mathbb{R}})\). In particular, if the group \(\{T_ t\}\) acts in \(H^ p({\mathbb{D}})\) and is associated as in Forelli's theorem with a group of parabolic Möbius transformations of \({\mathbb{D}}\), then it can be analyzed by way of the translation group on \(H^ p({\mathbb{R}})\). The Stone-type spectral family of the latter is shown to be obtained by restriction of the M. Riesz projections to \(H^ p({\mathbb{R}})\). By this means a concrete description of the Stone-type spectral family for a parabolic isometric group on \(H^ p({\mathbb{D}})\) is obtained. A pleasant by-product of the parabolic case is absorption of the classical Paley-Wiener theorem for \(H^ p({\mathbb{R}})\), \(1<p\leq 2\), into the framework of the generalized Stone's theorem.