Abstract
We obtain a generalization of the Burns-Krantz rigidity theorem for holomorphic self-mappings of the unit disk in the spirit of the classical Schwarz-Pick Lemma and its continuous version due to L. Harris via the generation theory for one-parameter semigroups. In particular, we establish geometric and analytic criteria for a holomorphic function on the disk with a boundary null point to be a generator of a semigroup of linear fractional transformations in term of relations among three boundary derivatives of the function at this point.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abate, The infinitesimal generators of semigroups of holomorphic maps, Ann. Mat. Pura Appl. (4) 161 (1992), 167–180.
D. Aharonov, M. Elin, S. Reich and D. Shoikhet, Parametric representations of semi-complete vector fields on the unit balls in ℂ n and Hilbert space, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 (1999), 229–253.
E. Berkson and H. Porta, Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), 101–115.
P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125, no. 596 (1997).
F. Bracci, R. Tauraso and F. Vlacci, Identity principles for commuting holomorphic self-maps of the unit disc, J. Math. Anal. Appl. 270 (2002), 451–473.
F. Bracci, M. D. Contreras and S. Díaz-Madrigal, Infinitesimal generators associated with semigroups of linear fractional maps, J. Anal. Math. 102 (2007), 119–142.
D. M. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661–676.
M. D. Contreras and S. Díaz-Madrigal, Analytic flows on the unit disk: angular derivatives and boundary fixed points, Pacific J. Math. 222 (2005), 253–286.
M. D. Contreras, S. Díaz-Madrigal and Ch. Pommerenke, Second angular derivatives and parabolic iteration in the unit disk, Trans. Amer. Math. Soc., to appear.
C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.
A. Denjoy, Sur l’itération des fonctions analytiques, C. R. Acad. Sci. Paris 182 (1926), 255–257.
M. Elin, M. Levinshtein, S. Reich and D. Shoikhet, Commuting semigroups of holomorphic mappings, Math. Scand., to appear.
M. Elin, S. Reich and D. Shoikhet, A Julia-Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball, Israel J. Math. 164 (2008), 397–411.
M. Elin and D. Shoikhet, Dynamic extension of the Julia-Wolff-Carathéodory Theorem, Dynam. Systems Appl. 10 (2001), 421–437.
M. Elin and D. Shoikhet, Semigroups with boundary fixed points on the unit Hilbert ball and spirallike mappings, in Geometric Function Theory in Several Complex Variables, World Sci. Publishing, River Edge, NJ, 2004, pp. 82–117.
L. A. Harris, A continuous form of Schwarz’s Lemma in normed linear spaces, Pacific J. Math. 38 (1973), 635–639.
T. L. Kriete and B. D. MacCluer, A rigidity theorem for composition operators on certain Bergman spaces, Michigan Math. J. 42 (1995), 379–386.
S. Migliorini and F. Vlacci, A new rigidity result for holomorphic maps, Indag. Math. (N.S.) 13 (2002), 537–549.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
S. Reich and D. Shoikhet, Generation theory for semigroups of holomorphic mappings in Banach spaces, Abstr. Appl. Anal. 1 (1996), 1–44.
S. Reich and D. Shoikhet, Semigroups and generators on convex domains with the hyperbolic metric, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8 (1997), 231–250.
S. Reich and D. Shoikhet, The Denjoy-Wolff theorem, Ann. Univ. Mariae Curie-Skłodowska Sect. A 51 (1997), 219–240.
S. Reich and D. Shoikhet, Metric domains, holomorphic mappings and non-linear semigroups, Abstr. Appl. Anal. 3 (1998), 203–228.
J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, Berlin, 1993.
D. Shoikhet, Semigroups in Geometrical Function Theory, Kluwer, Dordrecht, 2001.
D. Shoikhet, Representations of holomorphic generators and distortion theorems for spirallike functions with respect to a boundary point, Int. J. Pure Appl. Math. 5 (2003), 335–361.
R. Tauraso, Commuting holomorphic maps of the unit disc, Ergodic Theory Dynam. Systems 24 (2004), 945–953.
R. Tauraso and F. Vlacci, Rigidity at the boundary for self maps of the disk, Complex Variables Theory Appl. 45 (2001), 151–165.
J. Wolff, Sur l’iteration des fonctions holomorphes dans une region, et dont les valeurs appartiennent a cette region, C. R. Acad. Sci. Paris 182 (1926), 42–43.
J. Wolff, Sur l’iteration des fonctions bornées, C. R. Acad. Sci. Paris 182 (1926), 200–201.
J. Wolff, Sur une généralisation d’un théorème de Schwarz, C. R. Acad. Sci. Paris 182 (1926), 918–920.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shoikhet, D. Another look at the Burns-Krantz theorem. J Anal Math 105, 19–42 (2008). https://doi.org/10.1007/s11854-008-0030-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-008-0030-8