Abstract
We establish a Julia-Carathéodory theorem and a boundary Schwarz-Wolff lemma for hyperbolically monotone mappings in the open unit ball of a complex Hilbert space.
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References
M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, 1989.
M. Abate, Angular derivatives in several complex variables, in Real Methods in Complex and CR Geometry, Lecture Notes in Mathematics, 1848, Springer, Berlin, 2004, pp. 1–47.
M. Abate and R. Tauraso, The Julia-Wolff-Carathéodory theorem(s), in Complex geometric analysis in Pohang (1997), Contemporary Mathematics, 222, 1999, pp. 161–172.
D. Aharonov, M. Elin, S. Reich and D. Shoikhet, Parametric representations of semicomplete vector fields on the unit balls in ℂ n and in Hilbert space, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 10 (1999), 229–253.
L. Aizenberg and D. Shoikhet, Boundary behavior of semigroups of holomorphic mappings on the unit ball in ℂ n, Complex Variables. Theory and Application. An International Journal 47 (2002), 109–121.
F. Bracci, M. D. Contreras and S. Díaz-Madrigal, Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains, preprint, arXiv.math.CV/0607670, 2006.
M. D. Contreras, S. Díaz-Madrigal and Ch. Pommerenke, On boundary critical points for semigroups of analytic functions, Mathematica Scandinavia 98 (2006), 125–142.
C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.
S. Dineen, The Schwarz Lemma, Clarendon Press, Oxford, 1989.
M. Elin, S. Reich and D. Shoikhet, Asymptotic behavior of semigroups of ρ-nonexpansive and holomorphic mappings on the Hilbert ball, Annali di Matematica Pura ed Applicata. Series IV 181 (2002), 501–526.
M. Elin and D. Shoikhet, Dynamic extension of the Julia-Wolff-Carathéodory theorem, Dynamic Systems and Applications 10 (2001), 421–437.
M. Elin and D. Shoikhet, Semigroups of holomorphic mappings with boundary fixed points and spirallike mappings, in Geometric Function Theory in Several Complex Variables, World Sci. Publ., River Edge, NJ, 2004, pp. 82–117.
T. Franzoni and E. Vesentini, Holomorphic Maps and Invariant Distances, North-Holland, Amsterdam, 1980.
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984.
L. A. Harris, S. Reich and D. Shoikhet, Dissipative holomorphic functions, Bloch radii, and the Schwarz lemma, Journal d’Analyse Mathematique 82 (2000), 221–232.
E. Kopecká and S. Reich, Hyperbolic monotonicity in the Hilbert ball, Fixed Point Theory and Applications, 2006, Article ID 78104, pp. 1–15.
R. H. Martin, Jr., Differential equations on closed subsets of a Banach space, Transactions of the American Mathematical Society 179 (1973), 399–414.
Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer, Berlin, 1992.
S. Reich, On fixed point theorems obtained from existence theorems for differential equations, Journal of Mathematical Analysis and Applications 54 (1976), 26–36.
S. Reich and D. Shoikhet, Generation theory for semigroups of holomorphic mappings in Banach spaces, Abstract and Applied Analysis 1 (1996), 1–44.
S. Reich and D. Shoikhet, Semigroups and generators on convex domains with the hyperbolic metric, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni 8 (1997), pp. 231–250.
S. Reich and D. Shoikhet, The Denjoy-Wolff theorem, Encyclopaedia of Mathematics, Supplement III, Kluwer Academic Publishers, Dordrecht, 2001, pp. 121–123.
S. Reich and D. Shoikhet, Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces, Imperial College Press, London, 2005.
W. Rudin, Function Theory in the Unit Ball of ℂ n, Springer, Berlin, 1980.
J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, Berlin, 1993.
D. Shoikhet, Semigroups in Geometrical Function Theory, Kluwer, Dordrecht, 2001.
K. Yosida, Functional Analysis, Springer, Berlin, 1980.
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The second author was partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund-B. and G. Greenberg Research Fund (Ottawa).
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Elin, M., Reich, S. & Shoikhet, D. A Julia-Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball. Isr. J. Math. 164, 397–411 (2008). https://doi.org/10.1007/s11856-008-0037-y
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DOI: https://doi.org/10.1007/s11856-008-0037-y