Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T12:33:31.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation numbers of composition operators on the Hardy and Bergman spaces of the ball and of the polydisk

Published online by Cambridge University Press:  13 March 2017

FRÉDÉRIC BAYART ,
DANIEL LI ,
HERVÉ QUEFFÉLEC  and
LUIS RODRÍGUEZ–PIAZZA
FRÉDÉRIC BAYART
Affiliation:
Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont–Ferrand, France. e-mail: [email protected]
DANIEL LI
Affiliation:
Univ. Artois, Laboratoire de Mathématiques de Lens (LML) EA 2462, & Fédération CNRS Nord–Pas–de–Calais FR 2956, Faculté Jean Perrin, Rue Jean Souvraz, S.P. 18, F-62 300 Lens, France. e-mail: [email protected]
HERVÉ QUEFFÉLEC
Affiliation:
Univ. Lille Nord de France, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération CNRS Nord–Pas–de–Calais FR 2956, F-59 655 Villeneuve d'ascq Cedex, France. e-mail: [email protected]
LUIS RODRÍGUEZ–PIAZZA
Affiliation:
Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Apartado de Correos 1160, 41 080 Sevilla, Spain. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give general estimates for the approximation numbers of composition operators on the Hardy space on the ball Bd and the polydisk ${\mathbb D}$d and of composition operators on the Bergman space on the polydisk.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 1 , July 2018 , pp. 69 - 91
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Albiac, F. and Kalton, N.J.. Topics in Banach space theory. Graduate Texts in Math., 233. (Springer, New York, 2006).Google Scholar
[2] Bayart, F.. Composition operators on the polydisc induced by affine maps. J. Funct. Anal. 260 (2011), 1969–2003.CrossRefGoogle Scholar
[3] Bayart, F. and Brevig, O.. Compact composition operators with non-linear symbols on the H 2 space of Dirichlet series. Pacific J. Math., to appear (arXiv:1505.02944).Google Scholar
[4] Berndtsson, B.. Interpolating sequences for H ∞ in the ball. Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 1, 1–10.Google Scholar
[5] Berndtsson, B., Chang, S.-Y. and Lin, K.-C.. Interpolating sequences in the polydisc. Trans. Amer. Math. Soc. 302 (1987), 161–169.Google Scholar
[6] Carl, B. and Stephani, I.. Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, 98. (Cambridge University Press, Cambridge, 1990).Google Scholar
[7] Cima, J. A., Stanton, C. S. and Wogen, W. R.. On boundedness of composition operators on H 2(B 2). Proc. Amer. Math. Soc. 91 (1984), no. 2, 217–222.Google Scholar
[8] Cima, J. A. and Wogen, W. R.. Unbounded composition operators on H 2 (B 2). Proc. Amer. Math. Soc. 99 (1987), no. 3, 477–483.Google Scholar
[9] Clahane, D. D.. Spectra of compact composition operators over bounded symmetric domains. Integral Equations Operator Theory 51 (2005), no. 1, 41–56.Google Scholar
[10] Clerc, J.-L.. Geometry of the Shilov boundary of a bounded symmetric domain. J. Geom. Symmetry Phys. 13 (2009), 25–74.Google Scholar
[11] Clerc, J.-L.. Geometry of the Shilov boundary of a bounded symmetric domain. Geometry, integrability and quantization (Avangard Prima, Sofia, 2009), 11–55.Google Scholar
[12] Cowen, C. and MacCluer, B.. Composition Operators on Spaces of Analytic Functions. Stud. Adv. Math. (CRC Press, 1994).Google Scholar
[13] Finet, C., Queffélec, H. and Volberg, A.. Compactness of composition operators on a Hilbert space of Dirichlet series. J. Funct. Anal. 211 (2004), no. 2, 271–287.Google Scholar
[14] Garnett, J.. Bounded Analytic Functions, Revised First Edition (Springer, 2007).Google Scholar
[15] Guichardet, A.. Leçons sur Certaines Algèbres Topologiques (Gordon & Breach, Paris-London-New York, distributed by Editeur, Dunod, 1967).Google Scholar
[16] Hahn, K. T. and Mitchell, J.. H p spaces on bounded symmetric domains. Trans. Amer. Math. Soc. 146 (1969), 521–531.Google Scholar
[17] Hahn, K. T. and Mitchell, J.. H p spaces on bounded symmetric domains. Ann. Polon. Math. 28 (1973), 89–95.Google Scholar
[18] Hastings, W.. A Carleson measure theorem for Bergman spaces. Proc. Amer. Math. Soc. 52 (1975), 237–241.Google Scholar
[19] Krantz, S. G.. Function theory of several complex variables. Pure and Applied Mathematics. (A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1982).Google Scholar
[20] Lefèvre, P., Li, D., Queffélec, H. and Rodríguez–Piazza, L.. Some revisited results about composition operators on Hardy spaces. Rev. Mat. Iberoam. 28 (2012), No. 1, 57–76.Google Scholar
[21] Lefèvre, P., Li, D., Queffélec, H. and Rodríguez–Piazza, L.. Compact composition operators on Bergman–Orlicz spaces. Trans. Amer. Math. Soc. 365 (2013), no. 8, 3943–3970.Google Scholar
[22] Li, D. and Queffélec, H.. Introduction à l'étude des espaces de Banach. Analyse et Probabilités, Cours Spécialisés 12. (Société Mathématique de France, Paris, 2004).Google Scholar
[23] Li, D., Queffélec, H. and Rodríguez–Piazza, L.. On approximation numbers of composition operators. J. Approx. Theory 164 (2012), no. 4, 431–459.Google Scholar
[24] Li, D., Queffélec, H. and Rodríguez–Piazza, L.. Estimates for approximation numbers of some classes of composition operators on the Hardy space. Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 2, 547–564.Google Scholar
[25] Li, D., Queffélec, H. and Rodríguez–Piazza, L.. A spectral radius type formula for approximation numbers of composition operators. J. Funct. Anal. 267 (2014), no. 12, 4753–4774.Google Scholar
[26] Li, D., Queffélec, H. and Rodríguez–Piazza, L.. Approximation numbers of composition operators on H p. Concrete Operators 2 (2015), 98–109.Google Scholar
[27] Li, D., Queffélec, H. and Rodríguez–Piazza, L.. Composition operators on the Hardy space of the infinite polydisk. in preparation.Google Scholar
[28] MacCluer, B.. Spectra of compact composition operators on H p (B N). Analysis 4 (1984), 87–103.CrossRefGoogle Scholar
[29] Pietsch, A.. s-numbers of operators in Banach spaces. Studia Math. LI (1974), 201–223.Google Scholar
[30] Rudin, W.. Function Theory in the Unit Ball of ${\mathbb C}$n. Second Edition. (Springer, 2008).Google Scholar
[31] Shapiro, J. H.. Composition operators and classical function theory. Universitext, Tracts in Mathematics. (Springer–Verlag, New–York, 1993).Google Scholar
[32] Vigué, J.-P.. Le groupe des automorphismes analytiques d'un domaine borné d'un espace de Banach complexe. Application aux domaines bornés symétriques. Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 2, 203–281.Google Scholar