Abstract
We find sufficient conditions for a self-map of the unit ball to converge uniformly under iteration to a fixed point or idempotent on the entire ball. Using these tools, we establish spectral containments for weighted composition operators on Hardy and Bergman spaces of the ball. When the compositional symbol is in the Schur–Agler class, we establish the spectral radii of these weighted composition operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
No empirical data was used during this research.
References
Cowen, C.C., Ko, E., Thompson, D., Tian, F.: Spectra of some weighted composition operators on \({H}^{2}\). Acta Sci. Math. Szeged 82, 221–234 (2016)
Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. Studies in Advanced Mathematics, CRC Press, Boca Raton (1995)
Cowen, C.C., MacCluer, B.D.: Linear fractional maps of the ball and their composition operators. Acta Sci. Math. Szeged 66(1–2), 351–376 (2000)
Hyvärinen, O., Nieminen, I.: Essential spectra of weighted composition operators with hyperbolic symbols. Concr. Oper. 2(1), 66 (2015)
Jury, M.T.: Norms and spectral radii of linear fractional composition operators on the ball. J. Funct. Anal. 254(9), 2387–2400 (2008)
Jury, M.T.: Valiron’s theorem in the unit ball and spectra of composition operators. J. Math. Anal. Appl. 368(2), 482–490 (2010)
MacCluer, B.D.: Iterates of holomorphic self-maps of the unit ball in \({\mathbb{C} }^{N}\). Mich. Math. J. 30(1), 97–106 (1983)
Narasimhan, R.: Several Complex Variables. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1971)
Rudin, W.: Function Theory in the Unit Ball of \({ C}^{n}\). Fundamental Principles of Mathematical Science, vol. 241. Springer, New York (1980)
Shapiro, J.H.: Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer, New York (1993)
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)
Acknowledgements
Authors Kaschner, Makdad, Rempfer, Thompson, and Winters were supported by an NSF CURM grant and are grateful to the directors of CURM (Kathryn Leonard, Maria Mercedes Franco) for their counsel. We would also like to thank Michael Jury for his helpful advice.
Author information
Authors and Affiliations
Corresponding author
Additional information
Authors Kaschner, Makdad, Rempfer, Thompson, and Winters were supported by an NSF-CURM Grant.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kaschner, S., Le, T., Makdad, C. et al. Spectra of some weighted composition operators on the ball. Acta Sci. Math. (Szeged) 89, 373–387 (2023). https://doi.org/10.1007/s44146-023-00089-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s44146-023-00089-4