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Title

Spectral decompositions in Banach spaces and the Hilbert transform

Authors 1

Affiliations

  1. Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland

Abstract

This paper gives a survey of some recent developments in the spectral theory of linear operators on Banach spaces in which the Hilbert transform and its abstract analogues play a fundamental role.

Bibliography

  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.
  2. E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory 13 (1985), 33-47.
  3. E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations Operator Theory 9 (1986), 767-789.
  4. E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170.
  5. E. Berkson and T. A. Gillespie, The spectral decomposition of weighted shifts and the Ap condition, Colloq. Math. 60/61 (1990), 507-518.
  6. E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math. 112 (1994), 13-49.
  7. 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.
  8. E. Berkson, T. A. Gillespie and P. S. Muhly, Generalized analyticity in UMD spaces, Ark. Mat. 27 (1989), 1-14.
  9. J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, ibid. 21 (1983), 163-168.
  10. J. Bourgain, Vector-valued singular integrals and the H1-BMO duality, in: Probability Theory and Harmonic Analysis, J.-A. Chao and W. A. Woyczyński (eds.), Monographs and Textbooks in Pure and Appl. Math. 98, Marcel Dekker, New York, 1986, 1-19.
  11. D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in: Proc. Conf. in honor of A. Zygmund (Chicago, 1981), W. Beckner et al. (eds.), Wadsworth, Belmont, Calif., 1983, 270-286.
  12. M. L. Cartwright, Manuscripts of Hardy, Littlewood, Marcel Riesz and Titchmarsh, Bull. London Math. Soc. 14 (1982), 472-532.
  13. R. R. Coifman and G. Weiss, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, 1977.
  14. I. Doust, Well-bounded operators and the geometry of Banach spaces, Thesis, University of Edinburgh, 1988.
  15. I. Doust and B. Z. Qiu, The spectral theorem for well-bounded operators, J. Austral. Math. Soc. Ser. A 54 (1993), 334-351.
  16. H. R. Dowson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, London, 1978.
  17. H. R. Dowson and P. G. Spain, An example in the theory of well-bounded operators, Proc. Amer. Math. Soc. 32 (1972), 205-208.
  18. T. A. Gillespie, Logarithms of Lp translations, Indiana Univ. Math. J. 24 (1975), 1037-1045.
  19. T. A. Gillespie, A spectral theorem for Lp translations, J. London Math. Soc. (2) 11 (1975), 499-508.
  20. R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251.
  21. J. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1960), 334-343.
  22. J. R. Ringrose, On well-bounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613-638.
  23. D. R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. 1 (1960), 319-333.
  24. S. B. Stečkin, On bilinear forms, Dokl. Akad. Nauk SSSR 71 (1950), 237-240 (in Russian).
  25. E. C. Titchmarsh, Reciprocal formulae involving series and integrals, Math. Z. 25 (1926), 321-347.
  26. A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, Cambridge, 1959.
Banach Center Publications
1997/38/1
Export to PBN, Opens in new tab
Pages:
105-118
Main language of publication
English
Published
1997
Exact and natural sciences