Spectral families of projections in Hardy spaces
type: Item
instance of: scholarly article
title: Spectral families of projections in Hardy spaces
zbMATH Open document ID: 0554.47021
DOI: 10.1016/0022-1236(85)90049-7
author: Earl Berkson
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 3883064
zbMATH Keywords: Strongly continuous one-parameter groups of isometries
zbMATH Keywords: reflexive Hardy spaces
zbMATH Keywords: Stone's theorem for unitary groups
zbMATH Keywords: spectral decomposition
zbMATH Keywords: projection-valued function
zbMATH Keywords: spectral family
zbMATH Keywords: Forelli's theorem
zbMATH Keywords: parabolic Möbius transformations
zbMATH Keywords: translation group
zbMATH Keywords: Paley-Wiener theorem
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90049-7
OpenAlex ID: W2018845154
cites work: On the Convergence of Ordered Sets of Projections
cites work: Spectral Families of Projections, Semigroups, and Differential Operators
cites work: On Uniquely Decomposable Well-Bounded Operators
cites work: Mobius Transformations of the Disc and One-Parameter Groups of Isometries of H p
cites work: Hermitian Operators and One-Parameter Groups of Isometries in Hardy Spaces
cites work: Q4164422
cites work: Q5523220
cites work: Q3915429
cites work: The Isometries of Hp
cites work: A Spectral Theorem for Lp Translations
cites work: Q3244654
cites work: On the Theory of Fourier Transforms
cites work: Q5727906
cites work: Q5556050
cites work: On Well-Bounded Operators. II
cites work: A remark on the Volterra operator
cites work: Conditionally convergent spectral expansions
cites work: Q5607484
description: scientific article; zbMATH DE number 3883064