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.
@article{bwmeta1.element.bwnjournal-article-bcpv38i1p105bwm,
author = {Gillespie, T.},
title = {Spectral decompositions in Banach spaces and the Hilbert transform},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {105-118},
zbl = {0891.47022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p105bwm}
}
Gillespie, T. Spectral decompositions in Banach spaces and the Hilbert transform. Banach Center Publications, Tome 38 (1997) pp. 105-118. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv38i1p105bwm/
[000] [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. | Zbl 0509.47028
[001] [2] E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory 13 (1985), 33-47. | Zbl 0566.47011
[002] [3] E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations Operator Theory 9 (1986), 767-789. | Zbl 0607.47026
[003] [4] E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. | Zbl 0607.47027
[004] [5] E. Berkson and T. A. Gillespie, The spectral decomposition of weighted shifts and the condition, Colloq. Math. 60/61 (1990), 507-518. | Zbl 0819.47038
[005] [6] E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math. 112 (1994), 13-49. | Zbl 0823.42004
[006] [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. | Zbl 0609.47042
[007] [8] E. Berkson, T. A. Gillespie and P. S. Muhly, Generalized analyticity in UMD spaces, Ark. Mat. 27 (1989), 1-14.
[008] [9] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, ibid. 21 (1983), 163-168. | Zbl 0533.46008
[009] [10] J. Bourgain, Vector-valued singular integrals and the -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.
[010] [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.
[011] [12] M. L. Cartwright, Manuscripts of Hardy, Littlewood, Marcel Riesz and Titchmarsh, Bull. London Math. Soc. 14 (1982), 472-532.
[012] [13] R. R. Coifman and G. Weiss, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, 1977. | Zbl 0371.43009
[013] [14] I. Doust, Well-bounded operators and the geometry of Banach spaces, Thesis, University of Edinburgh, 1988.
[014] [15] I. Doust and B. Z. Qiu, The spectral theorem for well-bounded operators, J. Austral. Math. Soc. Ser. A 54 (1993), 334-351. | Zbl 0801.47022
[015] [16] H. R. Dowson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, London, 1978. | Zbl 0384.47001
[016] [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. | Zbl 0207.13705
[017] [18] T. A. Gillespie, Logarithms of translations, Indiana Univ. Math. J. 24 (1975), 1037-1045. | Zbl 0295.43009
[018] [19] T. A. Gillespie, A spectral theorem for translations, J. London Math. Soc. (2) 11 (1975), 499-508.
[019] [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. | Zbl 0262.44004
[020] [21] J. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1960), 334-343. | Zbl 0104.08902
[021] [22] J. R. Ringrose, On well-bounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613-638. | Zbl 0121.33403
[022] [23] D. R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. 1 (1960), 319-333. | Zbl 0104.08901
[023] [24] S. B. Stečkin, On bilinear forms, Dokl. Akad. Nauk SSSR 71 (1950), 237-240 (in Russian).
[024] [25] E. C. Titchmarsh, Reciprocal formulae involving series and integrals, Math. Z. 25 (1926), 321-347. | Zbl 52.0213.03
[025] [26] A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, Cambridge, 1959. | Zbl 0085.05601