Reich, Shoikhet: Semigroups and generators on convex domains with the hyperbolic metric

Reich, Simeon and Shoikhet, David:
Semigroups and generators on convex domains with the hyperbolic metric (Semigruppi e generatori su domini convessi con metrica iperbolica)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni Serie 9 8 (1997), fasc. n.4, p. 231-250, (English)
pdf (2.27 MB), djvu (424 Kb). | MR1631605 | Zbl 0905.47056

Sunto

Sia \( D \) un dominio in uno spazio di Banach complesso \( X \) e sia \( \rho \) una pseudometrica assegnata a \( D \) da un sistema di Schwarz-Pick. Nella prima parte del lavoro si stabiliscono alcuni criteri affinché una applicazione \( f : D \rightarrow X \) sia un generatore di un semigruppo \( \rho \)-non espansivo su \( D \). Nella seconda parte si suppone che sia \( X = H \), spazio di Hilbert complesso, che \( D = B \) disco unitario aperto di \( H \) e che sia \( \rho \) la metrica iperbolica su \( B \). Si introduce la nozione di applicazione \( \rho \)-monotona e si ottengono semplici caratterizzazioni di generatori di semigruppi di applicazioni olomorfe di \( B \) in sé.

Referenze Bibliografiche

[1] M. ABATE, Horospheres and iterates of holomorphic maps. Math. Z., 198, 1988, 225-238. | fulltext EuDML | fulltext (doi) | MR 939538 | Zbl 0628.32035

[2] M. ABATE, The infinitesimal generators of semigroups of holomorphic maps. Ann. Mat. Pura Appl., 161, 1992, 167-180. | fulltext (doi) | MR 1174816 | Zbl 0758.32013

[3] M. ABATE - J. P. VIGUÉ, Common fixed points in hyperbolic Riemann surfaces and convex domains. Proc. Amer. Math. Soc., 112, 1991, 503-512. | fulltext (doi) | MR 1065938 | Zbl 0724.32012

[4] L. AIZENBERG - S. REICH - D. SHOIKHET, One-sided estimates for the existence of null points of holomorphic mappings in Banach spaces. J. Math. Anal. Appl., 203, 1996, 38-54. | fulltext (doi) | MR 1412480 | Zbl 0869.46025

[5] J. ARAZY, An application of infinite dimensional holomorphy to the geometry of Banach spaces. Lecture Notes in Math., vol. 1267, Springer, Berlin 1987, 122-150. | fulltext (doi) | MR 907690 | Zbl 0622.46012

[6] V. BARBU, Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden 1976. | MR 390843 | Zbl 0328.47035

[7] E. BERKSON - H. PORTA, Semigroups of analytic functions and composition operators. Michigan Math. J., 25, 1978, 101-115. | fulltext mini-dml | MR 480965 | Zbl 0382.47017

[8] H. BRÉZIS, Opérateurs Maximaux Monotones. North Holland, Amsterdam 1973.

[9] H. CARTAN, Sur les rétractions d'une variété. C. R. Acad. Sci. Paris, 303, 1986, 715-716. | MR 870703 | Zbl 0609.32021

[10] P. CHERNOFF - J. E. MARSDEN, On continuity and smoothness of group actions. Bull. Amer. Math. Soc., 76, 1970, 1044-1049. | fulltext mini-dml | MR 265510 | Zbl 0202.23202

[11] M. G. CRANDALL - T. M. LIGGETT, Generation of semigroups of nonlinear transformations on general Banach spaces. Amer. J. Math., 93, 1971, 265-298. | MR 287357 | Zbl 0226.47038

[12] S. DINEEN, The Schwarz Lemma. Clarendon Press, Oxford 1989. | MR 1033739 | Zbl 0708.46046

[13] S. DINEEN - P. M. TIMONEY - J. P. VIGUÉ, Pseudodistances invariantes sur les domaines d'un espace localement convexe. Ann. Scuola Norm. Sup. Pisa, 12, 1985, 515-529. | fulltext EuDML | fulltext mini-dml | MR 848840 | Zbl 0603.46052

[14] C. J. EARLE - R. S. HAMILTON, A fixed point theorem for holomorphic mappings. Proc. Symp. Pure Math., vol. 16, Amer. Math. Soc., Providence, RI, 1970, 61-65. | MR 266009 | Zbl 0205.14702

[15] T. FRANZONI - E. VESENTINI, Holomorphic Maps and Invariant Distances. North Holland, Amsterdam 1980. | MR 563329 | Zbl 0447.46040

[16] I. I. GIKHMAN - A. V. SKOROKHOD, Theory of Random Processes. Nauka, Moscow 1973. | MR 341540 | Zbl 0348.60042

[17] K. GOEBEL - S. REICH, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Dekker, New York-Basel 1984. | MR 744194 | Zbl 0537.46001

[18] L. A. HARRIS, Schwarz-Pick systems of pseudometrics for domains in normed linear spaces. Advances in Holomorphy, North Holland, Amsterdam 1979, 345-406. | MR 520667 | Zbl 0409.46053

[19] T. E. HARRIS, The Theory of Branching Processes. Springer, Berlin 1963. | MR 163361 | Zbl 0117.13002

[20] U. HELMKE - B. MOORE, Optimization and Dynamical Systems. Springer, London 1994. | fulltext (doi) | MR 1299725 | Zbl 0943.93001

[21] M. E. JACOBSON, Computation of extinction probabilities for the Bellman-Harris branching process. Math. Biosciences, 77, 1985, 173-177. | fulltext (doi) | MR 820410 | Zbl 0575.60085

[22] V. KHATSKEVICH - S. REICH - D. SHOIKHET, Ergodic type theorems for nonlinear semigroups with holomorphic generators. Recent Developments in Evolution Equations, Pitman Research Notes in Math., vol. 324, 1995, 191-200. | MR 1417074 | Zbl 0863.47053

[23] V. KHATSKEVICH - S. REICH - D. SHOIKHET, Global implicit function and fixed point theorems for holomorphic mappings and semigroups. Complex Variables, 28, 1996, 347-356. | MR 1700203 | Zbl 0843.58007

[24] M. A. KRASNOSELSKI - P. P. ZABREIKO, Geometrical Methods of Nonlinear Analysis. Springer, Berlin 1984. | fulltext (doi) | MR 736839

[25] T. KUCZUMOW - A. STACHURA, Iterates of holomorphic and \( k_{D} \)-nonexpansive mappings in convex domains in \( \mathbf{C}^{n} \). Advances Math., 81, 1990, 90-98. | fulltext (doi) | MR 1051224 | Zbl 0726.32016

[26] P. MAZET - J. P. VIGUÉ, Points fixes d'une application holomorphe d'un domaine borné dans lui-même. Acta Math., 166, 1991, 1-26. | fulltext (doi) | MR 1088981 | Zbl 0733.32020

[27] W. RUDIN, The fixed point sets of some holomorphic maps. Bull. Malaysian Math. Soc., 1, 1978, 25-28. | MR 506535 | Zbl 0413.32012

[28] B. A. SEVASTYANOV, Branching Processes. Nauka, Moscow 1971. | MR 345229

[29] I. SHAFRIR, Coaccretive operators and firmly nonexpansive mappings in the Hilbert ball. Nonlinear Analysis, 18, 1992, 637-648. | fulltext (doi) | MR 1157564 | Zbl 0752.47018

[30] H. UPMEIER, Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics. CBMS - NSF Regional Conference Series in Math., AMS, Providence 1987. | MR 874756 | Zbl 0608.17013

[31] E. VESENTINI, Semigroups of holomorphic isometries. Advances Math., 65, 1987, 272-306. | fulltext (doi) | MR 904726 | Zbl 0642.47035

[32] E. VESENTINI, Krein spaces and holomorphic isometries of Cartan domains. In: S. COEN (ed.), Geometry and Complex Variables. Dekker, New York 1991, 409-413. | MR 1151658 | Zbl 0829.47029

[33] E. VESENTINI, Semigroups of holomorphic isometries. In: S. COEN (ed.), Complex Potential Theory. Kluwer, Dordrecht 1994, 475-548. | MR 1332968 | Zbl 0802.46058

[34] K. YOSIDA, Functional Analysis. Springer, Berlin 1968.

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche