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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
DOI: https://doi.org/10.1090/S0002-9947-1989-0948192-X
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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  • [1] H. Benzinger, E. Berkson, and T. A. Gillespie, Spectral families of projections, semigroups and differential operators, Trans. Amer. Math. Soc. 275 (1983), 431-475. MR 682713 (84b:47038)
  • [2] E. Berkson, Spectral families of projections in Hardy spaces, J. Funct. Anal. 60 (1985), 146-167. MR 777235 (86c:47039)
  • [3] E. Berkson and T. A. Gillespie, The generalized M. Riesz theorem and transference, Pacific J. Math. 120 (1985), 279-288. MR 810771 (87h:43004)
  • [4] -, Stečkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. MR 870759 (87m:47082)
  • [5] E. Berkson. T. A. Gillespie, and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3) 53 (1986), 489-517. MR 868456 (88e:47036)
  • [6] E. Berkson, R. Kaufman, and H. Porta, Màbius transformations of the disk and one-parameter groups of isometries of $ {H^p}$, Trans. Amer. Math. Soc. 199 (1974), 223-239. MR 0361923 (50:14365)
  • [7] E. Berkson and H. Porta, Hermitian operators and one-parameter groups of isometries in Hardy space, Trans. Amer. Math. Soc. 185 (1973), 331-344. MR 0338833 (49:3597)
  • [8] -, One parameter groups of isometries on Hardy spaces of the torus, Trans. Amer. Math. Soc. 220 (1976), 373-391. MR 0417855 (54:5903)
  • [9] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-spac-evalued functions, Proc. Conf. Harmonic Analysis in Honor of Antoni Zygmund (Chigago, Ill., 1981) (W. Beckner et al., eds.), Wadsworth, Belmont, Calif., 1983. MR 730072 (85i:42020)
  • [10] N. Dunford and J. T. Schwartz, Linear operators. I. General theory, Pure and Appl. Math., Vol. 7, Interscience, New York, 1958. MR 0117523 (22:8302)
  • [11] H. R. Dowson, Spectral theory of linear operators, London Math. Soc. Monographs, No. 12, Academic Press, New York, 1978. MR 511427 (80c:47022)
  • [12] P. L. Duren, Theory of $ {H^p}$ spaces, Pure and Appl. Math., Vol. 38, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [13] F. Forelli, The isometries of $ {H^p}$, Canad. J. Math. 16 (1964), 721-728. MR 0169081 (29:6336)
  • [14] J. B. Garnett, Bounded analytic functions, Pure and Appl. Math., Vol. 96, Academic Press, New York, 1981. MR 628971 (83g:30037)
  • [15] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 0133008 (24:A2844)
  • [16] D. Karayannakis, Spectral decompositions of isometries on Hardy spaces of the disk and the torus, Doctoral Dissertation, Univ. of Illinois at Urbana-Champaign, 1986.
  • [17] Y. Katznelson, An introduction to harmonic analysis, Dover, New York, 1976. MR 0422992 (54:10976)
  • [18] P. Koosis, Introduction to $ {H_p}$ spaces, London Math. Soc. Lecture Notes Series, no. 40, Cambridge Univ. Press, London and New York, 1980. MR 565451 (81c:30062)
  • [19] N. Lal and S. Merrill, Characterization of certain invariant subspaces of $ {H^p}$ and $ {L^p}$ spaces derived from logmodular algebras, Pacific J. Math. 30 (1969), 463-474. MR 0248526 (40:1778)
  • [20] F. Riesz and B. Sz-Nagy, Functional analysis, Ungar, New York, 1955. MR 0071727 (17:175i)
  • [21] Y. R. Ringrose, On well-bounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613-638. MR 0155185 (27:5124)
  • [22] D. Sarason, A remark on the Volterra operator, J. Math. Anal. Appl. 12 (1965), 244-246. MR 0192355 (33:580)

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Additional Information

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

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