Abstract.
We study generated semigroups of those self-mappings of the Hilbert ball which are non-expansive with respect to the hyperbolic metric. We find optimal convergence rates for such semigroups to interior stationary and boundary sink points. Since the hyperbolic metric is not defined on the boundary, the usual approach treats these two cases separately. In contrast with this practice, we use a special non-Euclidean “distance” (which induces the original topology) to present a unified theory. Our approach leads to new results even in the one-dimensional case. When the semigroups consist of holomorphic self-mappings, we obtain the rather unexpected phenomenon of universal rates of convergence of an exponential type. In particular, in the case of a boundary sink point we establish a continuous analog of the celebrated Julia–Wolff–Carathéodory theorem.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: January 3, 2001; in final form: November 28, 2001¶Published online: October 30, 2002
Rights and permissions
About this article
Cite this article
Elin, M., Reich, S. & Shoikhet, D. Asymptotic behavior of semigroups of ρ-non-expansive and holomorphic mappings on the Hilbert Ball. Ann. Mat. Pura Appl. IV. Ser. 181, 501–526 (2002). https://doi.org/10.1007/s10231-002-0052-2
Issue Date:
DOI: https://doi.org/10.1007/s10231-002-0052-2