Abstract
We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in ℂn with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc.
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References
John R. Akeroyd, Pratibha G. Ghatage, and Maria Tjani, Closed-range composition operators on A2 and the Bloch space, Integral Equations Operator Theory, 68 (2010), 503–517.
Timothy G. Clos, Mehmet Çelik, and Sönmez Şahutoğlu, Compactness of Hankel operators with symbols continuous on the closure of pseudoconvex domains, Integral Equations Operator Theory, 90 (2018), Art. 71, 14 pp.
Guangfu Cao, N. Elias, P. Ghatage, and Dahai Yu, Composition operators on a Banach space of analytic functions, Acta Math. Sinica (N.S.), 14 (1998), 201–208.
Timothy G. Clos, Compactness of Hankel Operators with Continuous Symbols on Domains in C2, PhD Thesis, The University of Toledo (2017).
Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press (Boca Raton, FL, 1995).
Željko Čucković and Sönmez Şahutoğlu, Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains, J. Funct. Anal., 256 (2009), 3730–3742.
Timothy G. Clos and Sönmez Şahutoğlu, Compactness of Hankel operators with continuous symbols, Complex Anal. Oper. Theory, 12 (2018), 365–376.
Željko Čucković and Ruhan Zhao, Essential norm estimates of weighted composition operators between Bergman spaces on strongly pseudoconvex domains, Math. Proc. Cambridge Philos. Soc., 142 (2007), 525–533.
Siqi Fu and Emil J. Straube, Compactness of the ∂̅-Neumann problem on convex domains, J. Funct. Anal., 159 (1998), 629–641.
Katherine Heller, Barbara D. MacCluer, and Rachel J. Weir, Compact differences of composition operators in several variables, Integral Equations Operator Theory, 69 (2011), 247–268.
Arthur Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883–890.
Joel H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer-Verlag (New York, 1993).
Emil J. Straube, Lectures on the l2-Sobolev Theory of the •̅-Neumann Problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS) (Zürich, 2010).
Akaki Tikaradze, Multiplication operators on the Bergman spaces of pseudoconvex domains, New York J. Math., 21 (2015), 1327–1345.
Kehe Zhu, Compact composition operators on Bergman spaces of the unit ball, Houston J. Math., 33 (2007), 273–283.
Acknowledgements
I thank Sönmez Şahutoğlu, Akaki Tikaradze, and Trieu Le for useful conversations and comments on a preliminary version of this manuscript. I also thank the anonymous referees for their useful suggestions.
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Clos, T.G. Compactness of composition operators on the Bergman spaces of convex domains and analytic discs. Anal Math 48, 29–37 (2022). https://doi.org/10.1007/s10476-021-0094-6
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DOI: https://doi.org/10.1007/s10476-021-0094-6