# Spectral decompositions in Banach spaces and the Hilbert transform

Banach Center Publications (1997)

- Volume: 38, Issue: 1, page 105-118
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topGillespie, T.. "Spectral decompositions in Banach spaces and the Hilbert transform." Banach Center Publications 38.1 (1997): 105-118. <http://eudml.org/doc/208623>.

@article{Gillespie1997,

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.},

author = {Gillespie, T.},

journal = {Banach Center Publications},

keywords = {spectral theory of linear operators on Banach spaces; Hilbert transform},

language = {eng},

number = {1},

pages = {105-118},

title = {Spectral decompositions in Banach spaces and the Hilbert transform},

url = {http://eudml.org/doc/208623},

volume = {38},

year = {1997},

}

TY - JOUR

AU - Gillespie, T.

TI - Spectral decompositions in Banach spaces and the Hilbert transform

JO - Banach Center Publications

PY - 1997

VL - 38

IS - 1

SP - 105

EP - 118

AB - 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.

LA - eng

KW - spectral theory of linear operators on Banach spaces; Hilbert transform

UR - http://eudml.org/doc/208623

ER -

## References

top- [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. Zbl0509.47028
- [2] E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory 13 (1985), 33-47. Zbl0566.47011
- [3] E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations Operator Theory 9 (1986), 767-789. Zbl0607.47026
- [4] E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Funct. Anal. 70 (1987), 140-170. Zbl0607.47027
- [5] E. Berkson and T. A. Gillespie, The spectral decomposition of weighted shifts and the ${A}_{p}$ condition, Colloq. Math. 60/61 (1990), 507-518. Zbl0819.47038
- [6] E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math. 112 (1994), 13-49. Zbl0823.42004
- [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. Zbl0609.47042
- [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. Zbl0533.46008
- [10] J. Bourgain, Vector-valued singular integrals and the ${H}^{1}$-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. Zbl0371.43009
- [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. Zbl0801.47022
- [16] H. R. Dowson, Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, London, 1978. Zbl0384.47001
- [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. Zbl0207.13705
- [18] T. A. Gillespie, Logarithms of ${L}^{p}$ translations, Indiana Univ. Math. J. 24 (1975), 1037-1045. Zbl0295.43009
- [19] T. A. Gillespie, A spectral theorem for ${L}^{p}$ 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. Zbl0262.44004
- [21] J. R. Ringrose, On well-bounded operators, J. Austral. Math. Soc. 1 (1960), 334-343. Zbl0104.08902
- [22] J. R. Ringrose, On well-bounded operators II, Proc. London Math. Soc. (3) 13 (1963), 613-638. Zbl0121.33403
- [23] D. R. Smart, Conditionally convergent spectral expansions, J. Austral. Math. Soc. 1 (1960), 319-333. Zbl0104.08901
- [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. Zbl52.0213.03
- [26] A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, Cambridge, 1959. Zbl0085.05601

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.