Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] E. Berkson and H. Porta, Hermitian operators and one-parameter groups of isometries in Hardy spaces, Trans. Amer. Math. Soc. 185 (1973), 331-344. MR 0338833 (49:3597)
  • [2] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [3] P. Duren, Theory of $ {H^p}$ spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 42 #3552. MR 0268655 (42:3552)
  • [4] F. Forelli, The isometries of $ {H^p}$, Canad. J. Math. 16 (1964), 721-728. MR 29 #6336. MR 0169081 (29:6336)
  • [5] R. Gabriel, Some results concerning the integrals of moduli of regular functions along certain curves, J. London Math. Soc. 2 (1927), 112-117.
  • [6] E. Hille and R. S. Phillips, Functional analysis and semi-groups, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 19, 664. MR 0089373 (19:664d)
  • [7] N. Levinson and R. M. Redheffer, Complex variables, Holden-Day, San Francisco, Calif., 1970. MR 42 #6193. MR 0271310 (42:6193)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47D10, 46E15

Retrieve articles in all journals with MSC: 47D10, 46E15


Additional Information

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

American Mathematical Society