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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral decompositions of one-parameter groups of isometries on Hardy spaces


Author: Dimitri Karayannakis
Journal: Trans. Amer. Math. Soc. 311 (1989), 147-166
MSC: Primary 47D05; Secondary 30D55, 42A45, 43A50, 46E15
MathSciNet review: 948192
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Abstract: Spectral decompositions of strongly continuous one-parameter groups of surjective isometries on Hardy spaces of the disk $ {\mathbf{D}}$ and the torus $ {{\mathbf{T}}^2}$ are examined; a concrete description of the (pointwise) action of these decompositions is presented, mainly in the parabolic case, leading to a complete description of the action of the partial sum-operators of M. Riesz when carried from $ {L^p}({\mathbf{R}})$ to $ {H^p}({\mathbf{D}})$, $ 1 < p \leq 2$. The (pointwise) action of the spectral decompositions of these isometric groups on $ {H^p}({{\mathbf{T}}^2})$, $ 1 < p < \infty $ is also examined and concrete descriptions are derived, mainly in the parabolic case.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0948192-X
PII: S 0002-9947(1989)0948192-X
Keywords: Isometry, group of isometries, Hardy space, spectral family, group of Màbius transformations, projection, residue theory
Article copyright: © Copyright 1989 American Mathematical Society