An example in holomorphic fixed point theory

Budzyńska, Monika

doi:10.1090/S0002-9939-03-06982-X

An example in holomorphic fixed point theory
HTML articles powered by AMS MathViewer

by Monika Budzyńska PDF

Proc. Amer. Math. Soc. 131 (2003), 2771-2777 Request permission

Abstract:

If $B$ is the open unit ball in the Cartesian product $l^2 \times l^2$ furnished with the $l^p$-norm $\|\cdot \|$, where $1 <p < \infty$ and $p \neq 2$, then a holomorphic self-mapping $f$ of $B$ has a fixed point if and only if $\sup _{n\in \mathbb {N}} \|f^n (x)\| <1$ for some $x\in B.$

References

M. Budzyńska & T. Kuczumow, A strict convexity of the Kobayashi distance, Fixed Point Theory and Applications, Vol. 4 (Ed. Y. J. Cho), Nova Science Publishers, Inc., to appear.
Seán Dineen, The Schwarz lemma, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR 1033739
Clifford J. Earle and Richard S. Hamilton, A fixed point theorem for holomorphic mappings, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 61–65. MR 0266009
Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206–208. MR 291917, DOI 10.1090/S0002-9904-1972-12918-5
Tullio Franzoni and Edoardo Vesentini, Holomorphic maps and invariant distances, Notas de Matemática [Mathematical Notes], vol. 69, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 563329
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
K. Goebel, T. Sekowski, and A. Stachura, Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal. 4 (1980), no. 5, 1011–1021. MR 586863, DOI 10.1016/0362-546X(80)90012-7
Lawrence A. Harris, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces, Advances in holomorphy (Proc. Sem. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) North-Holland Math. Stud., vol. 34, North-Holland, Amsterdam-New York, 1979, pp. 345–406. MR 520667
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
Wilhelm Kaup and Harald Upmeier, Banach spaces with biholomorphically equivalent unit balls are isomorphic, Proc. Amer. Math. Soc. 58 (1976), 129–133. MR 422704, DOI 10.1090/S0002-9939-1976-0422704-3
T. Kuczumow, Fixed points of holomorphic mappings in the Hilbert ball, Colloq. Math. 55 (1988), no. 1, 101–107. MR 964327, DOI 10.4064/cm-55-1-101-107
Tadeusz Kuczumow, Simeon Reich, and David Shoikhet, The existence and non-existence of common fixed points for commuting families of holomorphic mappings, Nonlinear Anal. 43 (2001), no. 1, 45–59. MR 1784444, DOI 10.1016/S0362-546X(99)00175-3
T. Kuczumow, S. Reich & D. Shoikhet, Fixed points of holomorphic mappings: a metric approach, Handbook of Metric Fixed Point Theory (Eds. W. A. Kirk and B. Sims), Kluwer Academic Publishers, 2001, 437-515.
Tadeusz Kuczumow, Simeon Reich, and Adam Stachura, Holomorphic retracts of the open unit ball in the $l_\infty$-product of Hilbert spaces, Recent advances on metric fixed point theory (Seville, 1995) Ciencias, vol. 48, Univ. Sevilla, Seville, 1996, pp. 99–110. MR 1440221
Tadeusz Kuczumow and Adam Stachura, Iterates of holomorphic and $k_D$-nonexpansive mappings in convex domains in $\textbf {C}^n$, Adv. Math. 81 (1990), no. 1, 90–98. MR 1051224, DOI 10.1016/0001-8708(90)90005-8
Simeon Reich, Averaged mappings in the Hilbert ball, J. Math. Anal. Appl. 109 (1985), no. 1, 199–206. MR 796053, DOI 10.1016/0022-247X(85)90187-8
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
Tadeusz Sękowski and Adam Stachura, Holomorphic non-equivalence of balls in Banach spaces $l_p$ and $L_2$ from the geometrical point of view, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 213–218. MR 1472592
José M. Isidro and Jean-Pierre Vigué, On the product property of the Carathéodory pseudodistance, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 1, 21–26 (English, with English and Italian summaries). MR 1797050
Jean-Pierre Vigué, Stricte convexité des domaines bornés et unicité des géodésiques complexes, Bull. Sci. Math. 125 (2001), no. 4, 297–310 (French, with English and French summaries). MR 1831983, DOI 10.1016/S0007-4497(00)01079-4