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Lacunarity on nonabelian groups and summing operators

Part of: Special classes of linear operators Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

P. G. Dodds
P. G. Dodds
Affiliation:
1/242 Sir Fred Schonell Drive St. Lucia QLD 4067Australia e-mail: lermer@mailbox.uq.edu.au
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Abstract

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In this note we investigate lacunarity or ‘thin’ subsets in the dual object of a compact group via different classes of summing operators between Banach spaces. In particular, we give characterisations of Sidon and ∧ (p) sets, 2 < p < ∞

MSC classification

Secondary: 47B10: Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) 42A55: Lacunary series of trigonometric and other functions; Riesz products
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 1 , August 2001 , pp. 71 - 79
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Baur, F., Banach operator ideals generated by orthonormal systems (Ph.D. Thesis, Universität Zürich, 1997).Google Scholar
[2]Baur, F., ‘∧(2)-systems and summing operators’, Arch. Math. 71 (1998), 465–471.CrossRefGoogle Scholar
[3]Baur, F., ‘Operator ideals, orthonormal systems and lacunary sets’, Math. Nachr. 197 (1999), 19–28.CrossRefGoogle Scholar
[4]Defant, A. and Floret, K., Tensor norms and operator ideals (North-Holland, Amsterdam, 1993).Google Scholar
[5]Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators (Cambridge Univ. Press, Cambridge, 1995).CrossRefGoogle Scholar
[6]Linde, W. and Pietsch, A., ‘Mappings of Gaussian cylindrical measures in Banach spaces’, Theory Probab. Appl. 19 (1974), 445–460.CrossRefGoogle Scholar
[7]Marcus, M. B. and Pisier, G., Random Fourier series with application to harmonic analysis, Ann. of Math. Studies 101 (Princeton Univ. Press, Princeton, 1981).Google Scholar
[8]Pietsch, A., Operator ideals (North-Holland, Amsterdam, 1980).Google Scholar
[9]Pisier, G., ‘Lesinégalités de Khinchine-Kahane d'après C. Borell’, in: Séminaire sur la Géométrie des Espaces de Banach 1977–1978, Exposé No VII (eds. Maurey, B. and Schwartz, L.) (Ecole Polytechnique, Centre de Mathématiques, 1978), pp. VII 1–VII 14.Google Scholar
[10]Price, J. F., ‘Non ci sono insiemi infiniti di dipo ∧(p) per SU (2)’, Boll. Un. Mat. Ital. 4 (1971), 879–881.Google Scholar
[11]Rider, D., ‘Central lacunary sets’, Monatsh. Math. 76 (1972), 328–338.CrossRefGoogle Scholar
[12]Rider, D., Norms of character and central ∧p sets for U(n), Lecture Notes in Math. 266 (Springer, Berlin, 1972).Google Scholar
[13]Rider, D., ‘Randomly continuous functions and Sidon sets’, Duke Math. J. 42 (1975), 759–764.CrossRefGoogle Scholar
[14]Rider, D., ‘SU(n) has no infinite local ∧p sets’, Boll. Un. Mat. Ital. 12 (1975), 155–160.Google Scholar
[15]Seigner, J., Über eine Klasse von Idealnormen, die mit Orthonormalsystemen gebildet sind (Ph.D. Thesis, Friedrich-Schiller-Universität, Jena, 1995).Google Scholar
[16]Vrem, R. C., ‘Lacunarity on compact hypergroups’, Math. Z. 164 (1978), 93–104.CrossRefGoogle Scholar
[17]Vrem, R. C., ‘Independent sets and lacunarity for hypergroups’, J. Austral. Math. Soc. (Series A) 50 (1991), 171–188.CrossRefGoogle Scholar