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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Möbius transformations of the disc and one-parameter groups of isometries of $ H\sp{p}$


Authors: Earl Berkson, Robert Kaufman and Horacio Porta
Journal: Trans. Amer. Math. Soc. 199 (1974), 223-239
MSC: Primary 47D10; Secondary 46E15
DOI: https://doi.org/10.1090/S0002-9947-1974-0361923-4
MathSciNet review: 0361923
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Abstract: Let $ \{ {T_t}\} $ be a strongly continuous one-parameter group of isometries in $ {H^p}(1 \leqslant p < \infty ,p \ne 2)$ with unbounded generator. There is a uniquely determined one-parameter group $ \{ {\phi _t}\} ,t \in {\mathbf{R}}$, of Möbius transformations of the (open) disc $ D$ corresponding to $ \{ {T_t}\} $ by way of Forelli's theorem. The interplay between $ \{ {T_t}\} $ and $ \{ {\phi _t}\} $ is studied, and the spectral properties of the generator $ A$ of $ \{ {T_t}\} $ are analyzed in this context. The nature of the set $ S$ of common fixed points of the functions $ {\phi _t}$ plays a crucial role in determining the behavior of $ A$. The spectrum of $ A$, which is a subset of $ i{\mathbf{R}}$, can be a discrete set, a translate of $ i{{\mathbf{R}}_ + }$ or of $ i{{\mathbf{R}}_ - }$, or all of $ i{\mathbf{R}}$. If $ S$ is not a doubleton subset of the unit circle, $ \{ {T_t}\} $ can be extended to a holomorphic semigroup of $ {H^p}$-operators, the semigroup being defined on a half-plane. The treatment of $ \{ {T_t}\} $ is facilitated by developing appropriate properties of one-parameter groups of Möbius transformations of $ D$. In particular, such groups are in one-to-one correspondence (via an initial-value problem) with the nonzero polynomials $ q$, of degree at most 2, such that Re$ [\bar zq(z)] = 0$ for all unimodular $ z$. A has an explicit description (in terms of the polynomial corresponding to $ \{ {\phi _t}\} $) as a differential operator.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0361923-4
Keywords: Isometry, $ {H^p}$, Möbius transformation, group, generator, spectrum
Article copyright: © Copyright 1974 American Mathematical Society