Abstract and Applied Analysis

Generation theory for semigroups of holomorphic mappings in Banach spaces

Simeon Reich and David Shoikhet

Full-text: Open access

Abstract

We study nonlinear semigroups of holomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog of the Hille exponential formula. We then apply our results to the null point theory of semi-plus complete vector fields. We study the structure of null point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of the implicit function theorem and a discussion of some open problems are also included.

Article information

Source
Abstr. Appl. Anal., Volume 1, Number 1 (1996), 1-44.

Dates
First available in Project Euclid: 7 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1049725990

Digital Object Identifier
doi:10.1155/S1085337596000012

Mathematical Reviews number (MathSciNet)
MR1390558

Zentralblatt MATH identifier
0945.46026

Subjects
Primary: 32H15 34G20
Secondary: 46G20 47H10 47H15 47H20

Keywords
Banach space Cauchy problem exponential formula holomorphic generator hyperbolic metric Lie generator nonlinear semigroup null point resolvent spectrum

Citation

Reich, Simeon; Shoikhet, David. Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstr. Appl. Anal. 1 (1996), no. 1, 1--44. doi:10.1155/S1085337596000012. https://projecteuclid.org/euclid.aaa/1049725990


Export citation

References

  • Abate, M., Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), 225--238.
  • Abate, M., The infinitesimal generators of semigroups of holomorphic maps, Ann. Mat. Pura Appl. (4) 161 (1992), 167--180.
  • Abd-Alla, M., L'ensemble des points fixes d'une application holomorphe dans un produit fini de boules-unités d'espaces de Hilbert et une sous-variété banachique complexe, Ann. Mat. Pura Appl. (4) 153 (1988), 63--75.
  • Abts, D., On injective holomorphic Fredholm mappings of index 0 in complex Banach spaces, Comment. Math. Univ. Carolinae 21 (1980), 513--525.
  • Aizenberg, L., Reich, S., and Shoikhet, D., One-sided estimates for the existence of null points of holomorphic mappings in Banach spaces, Technion Preprint Series No. MT-1015, 1995, J. Math. Anal. Appl. (to appear).
  • Angenent, S.B., Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh 115 (1990), 91--107.
  • Arazy, J., An application of infinite dimensional holomorphy to the geometry of Banach spaces, Lecture Notes in Math. 1267, Springer, Berlin, 1987, 122--150.
  • Azizov, T. Y., Khatskevich, V. and Shoikhet, D., On the number of fixed points of a holomorphism, Siberian Math. J. 31 (1990), 192--195.
  • Barbu, V., Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff, Leyden, 1976.
  • Brézis, H., Opérateurs Maximaux Monotones, North Holland, Amsterdam, 1973.
  • Brézis, H., and Pazy, A., Convergence and approximation of nonlinear operators in Banach spaces, J. Funct. Anal. 9 (1972), 63--64.
  • Bruck, R.E., Structure of the approximate fixed-point sets of nonexpansive mappings in general Banach spaces, Pitman Res. Notes Math. 252 (1991), 91--96.
  • Cartan, H., Calcul Différentiel, Hermann, Paris, 1967.
  • Chirka, E., Complex Analytic Sets, Nauka, Moscow, 1985.
  • Daletskii, Yu. L. and Krein, M. G., Stability of Solutions of Differential Equations in a Banach Space, Nauka, Moscow, 1970.
  • Dieudonné, J., Foundations of Modern Analysis, Academic Press, New York and London, 1960.
  • Dineen, S., The Schwarz Lemma, Clarendon Press, Oxford, 1989.
  • Dineen, S., Timoney, P.M. and Vigué, J.P., Pseudodistances invariantes sur les domaines d'un espace localement convexe, Ann. Scuola Norm. Sup. Pisa 12 (1985), 515--529.
  • Do Duc Thai, The fixed points of holomorphic maps on a convex domain, Ann. Polon. Math. 56 (1992), 143--148.
  • Dorroh, J. R. and Neuberger, J. W., Lie generators for semigroups of transformations on a Polish space, Electron. J. Differential Equations 1 (1993), 1--7.
  • Earle, C. J. and Hamilton, R. S., A fixed-point theorem for holomorphic mappings, Proc. Sympos. Pure Math. Vol. 16, Amer. Math. Soc., Providence, R. I., 1970, pp. 61--65.
  • Franzoni, T. and Vesentini, E., Holomorphic Maps and Invariant Distances, Amsterdam, North-Holland 1980.
  • Goebel, K. and Reich, S., Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
  • Goebel, K., Sekowski, T. and Stachura, A., Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal. 4 (1980), 1011--1021.
  • Harris, L. A., Schwarz's lemma in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014--1017.
  • Harris, L. A., A continuous form of Schwarz's lemma in normed linear spaces, Pacific J. Math. 38 (1971), 635--639.
  • Harris, L. A., The numerical range of holomorphic functions in Banach spaces, Amer. J. Math. 93 (1971), 1005--1019.
  • Harris, T. E., The Theory of Branching Processes, Springer, Berlin, 1963.
  • Hayden, T. L. and Suffridge, T. J., Biholomorphic maps in Hilbert space have a fixed point, Pacific J. Math. 38 (1971) 419--422.
  • Hayden, T. L. and Suffridge, T. J., Fixed points of holomorphic maps in Banach spaces, Proc. Amer. Math. Soc. 60 (1976), 95--105.
  • Heath, L. F. and Suffridge, T. J., Holomorphic retracts in complex $n$-space, Illinois J. Math. 25 (1981), 125--135.
  • Helmke, U. and Moore, J. B., Optimization and Dynamical Systems, Springer, London, 1994.
  • Hervé, M., Analyticity in Infinite Dimensional Spaces, De Gruyter, Berlin 1989.
  • Hille, E. and Phillips, R., Functional Analysis and Semigroups, Amer. Math. Soc., Providence, R. I., 1957.
  • Isidro, J. M. and Stacho, L. L., Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction, North-Holland, Amsterdam, 1984.
  • Kato, T., Perturbation Theory for Linear Operators, Springer, Berlin, 1966.
  • Khatskevich, V., Reich, S. and Shoikhet, D., Global implicit function and fixed point theorems for holomorphic mappings and semigroups, Complex Variables (to appear).
  • Khatskevich, V., Reich, S. and Shoikhet, D., Fixed point theorems for holomorphic mappings and operator theory in indefinite metric spaces, Integral Equations Operator Theory 22 (1995), 305--316.
  • Khatskevich, V., Reich, S. and Shoikhet, D., Ergodic type theorems for nonlinear semigroups with holomorphic generators, Recent Developments in Evolution Equations, Pitman Res. Notes Math. 324 (1995), 191-200.
  • Khatskevich, V. and Shoikhet, D., Fixed points of analytic operators in a Banach space and applications, Siberian Math. J. 25 (1984), 188-200.
  • Khatskevich V. and Shoikhet D., Differentiable Operators and Nonlinear Equations, Birkhäuser, Basel, 1994.
  • Khatskevich, V. and Shoikhet, D., One version of implicit function theorem for holomorphic mappings, C. R. Acad. Sci. Paris, 319 (1994), 599--604.
  • Khatskevich, V. and Shoikhet, D., Null-point sets of holomorphic generators of one-parameter semigroups, Dynam. Systems Appl. 4 (1995), 611 - 629.
  • Kobayashi, Y. and Oharu, S., Semigroups of locally Lipschitzian operators in Banach spaces, Hiroshima Math. J. 20 (1990), 573--611.
  • Krasnoselski, M.A., Vainikko, G.M., Zabreiko, P.P., Ruticki, Ya.B. and Stecenko, V.Ya., Approximate Solution of Operator Equations, Nauka, Moscow, 1969.
  • Krasnoselski, M. A. and Zabreiko, P. P., Geometric Methods of Nonlinear Analysis, Springer, Berlin, 1984.
  • Kuczumow, T. and Stachura, A., Convexity and fixed points of holomorphic mappings in Hilbert ball and polydisk, Bull. Polish Acad. Sci. 34 (1986), 189--193.
  • Kuczumow, T. and Stachura, A., Fixed points of holomorphic mappings in Cartesian products of $n$ unit Hilbert balls, Canad. Math. Bull. 29 (1986), 281--286.
  • Lempert, L., Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), 257--261.
  • Martin, R.H., Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc., 179 (1973), 399-414.
  • Mazet, P., Les points fixes d'une application holomorphe d'un domaine borné dans lui-même admettent une base de voisinages convexes stable, C. R. Acad. Sci. Paris 314 (1992), 197--199.
  • Mazet, P. and Vigué, J. P., Points fixes d'une application holomorphe d'un domaine borné dans lui-même, Acta Math. 166 (1991), 1--26.
  • Oharu, S. and Takahashi, T., Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups, Proc. Amer. Math. Soc. 100 (1987), 187--194.
  • Oharu, S. and Takahashi, T., Characterization of nonlinear semigroups associated with semilinear evolution equations, Trans. Amer. Math. Soc. 311 (1989), 593--619.
  • Pazy A., Semigroups of nonlinear contractions and their asymptotic behavior, Nonlinear Analysis and Mechanics, Pitman Res. Notes Math. 30 (1979), 36-134.
  • Reich, S., On fixed point theorems obtained from existence theorems for differential equations, J. Math. Anal. Appl. 54 (1976), 26--36.
  • Reich, S., Product formulas, nonlinear semigroups and accretive operators, J. Funct. Anal. 36 (1980), 147--168.
  • Reich, S., A nonlinear Hille-Yosida theorem in Banach spaces, J. Math. Anal. Appl. 84 (1981), 1--5.
  • Reich, S., The asymptotic behavior of a class of nonlinear semigroups in the Hilbert ball, J. Math. Anal. Appl. 157 (1991), 237--242.
  • Reich, S., Approximating fixed points of holomorphic mappings, Math. Japon. 37 (1992), 457--459.
  • Reich, S., and Shafrir, I., Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15 (1990), 537--558.
  • Reich, S. and Torrejon, R., Zeros of accretive operators, Comment. Math. Univ. Carolin. 21 (1980), 619--625.
  • Rudin, W., The fixed-point set of some holomorphic maps, Bull. Malaysian Math. Soc. 1 (1978), 25--28.
  • Sevastyanov, B. A., Branching Processes, Nauka, Moscow, 1971.
  • Shoikhet, D., Some properties of Fredholm mappings in Banach analytic manifolds, Integral Equations Operator Theory 16 (1993), 430--451.
  • Trenogin, V.A., Functional Analysis, Nauka, Moscow, 1980.
  • Upmeier H., Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, CBMS Regional Conf. Ser. in Math., 67, Amer. Math. Soc., Providence, R.I., 1986.
  • Vainberg, M. M. and Trenogin, V. A., Theory of Bifurcation of Solutions of Nonlinear Equations, Moscow, Nauka 1969.
  • Vesentini, E., Complex geodesics and holomorphic maps, Sympos. Math. 26 (1982), 211--230.
  • Vesentini, E., Iterates of holomorphic mappings, Uspekhi Mat. Nauk 40 (1985), 13--16.
  • Vesentini, E., Su un teorema di Wolff e Denjoy, Rend. Sem. Mat. Fis. Milano 53 (1983), 17--26 (1986).
  • Vesentini, E., Krein spaces and holomorphic isometries of Cartan domains, Geometry and Complex Variables, Marcel Dekker, 1991, pp. 409--443.
  • Vesentini, E., Semigroups of holomorphic isometries, Complex Potential Theory, Kluwer, Dordrecht, 1994, pp. 475--548.
  • Vigué, J. P., Points fixes d'applications holomorphes dans un produit fini de boules-unités d'espaces de Hilbert, Ann Mat. Pura Appl. (4) 137 (1984), 245--256.
  • Vigué, J. P., Sur les points fixes d'applications holomorphes, C. R. Acad. Sci. Paris 303 (1986), 927--930.
  • Vigué, J. P., Fixed points of holomorphic mappings in a bounded convex domain in $C^n$, Proc. Sympos. Pure Math. Vol. 52 (1991), Part 2, pp. 579--582.
  • Yosida, K., Functional Analysis, Springer, Berlin, 1965.