The theorems of Glicksberg and Hurwitz for holomorphic maps in complex Banach spaces

Kazimier Wzodarczyk

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993)

  • Volume: 4, Issue: 2, page 99-102
  • ISSN: 1120-6330

Abstract

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General versions of Glicksberg's theorem concerning zeros of holomorphic maps and of Hurwitz's theorem on sequences of analytic functions is extended to infinite dimensional Banach spaces.

How to cite

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Wzodarczyk, Kazimier. "The theorems of Glicksberg and Hurwitz for holomorphic maps in complex Banach spaces." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 4.2 (1993): 99-102. <http://eudml.org/doc/244156>.

@article{Wzodarczyk1993,
abstract = {General versions of Glicksberg's theorem concerning zeros of holomorphic maps and of Hurwitz's theorem on sequences of analytic functions is extended to infinite dimensional Banach spaces.},
author = {Wzodarczyk, Kazimier},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Fréchet holomorphic map; Local uniform convergence; Compact open topology; local uniform convergence; compact open topology; Hurwitz Theorem for Banach spaces; number of zeros of holomorphic maps to infinite- dimensional Banach spaces},
language = {eng},
month = {6},
number = {2},
pages = {99-102},
publisher = {Accademia Nazionale dei Lincei},
title = {The theorems of Glicksberg and Hurwitz for holomorphic maps in complex Banach spaces},
url = {http://eudml.org/doc/244156},
volume = {4},
year = {1993},
}

TY - JOUR
AU - Wzodarczyk, Kazimier
TI - The theorems of Glicksberg and Hurwitz for holomorphic maps in complex Banach spaces
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1993/6//
PB - Accademia Nazionale dei Lincei
VL - 4
IS - 2
SP - 99
EP - 102
AB - General versions of Glicksberg's theorem concerning zeros of holomorphic maps and of Hurwitz's theorem on sequences of analytic functions is extended to infinite dimensional Banach spaces.
LA - eng
KW - Fréchet holomorphic map; Local uniform convergence; Compact open topology; local uniform convergence; compact open topology; Hurwitz Theorem for Banach spaces; number of zeros of holomorphic maps to infinite- dimensional Banach spaces
UR - http://eudml.org/doc/244156
ER -

References

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  1. CONWAY, J. B., Functions of One Complex Variable. Graduate Texts in Math., 11, 2nd ed., Springer-Verlag, New York-Berlin-Heidelberg-Tokyo1978. Zbl0277.30001MR503901
  2. FRANZONI, T. - VESENTINI, E., Holomorphic Maps and Invariant Distances. North Holland Mathematics Studies, 40, Amsterdam-New York-Oxford1980. Zbl0447.46040MR563329
  3. GLICKSBERG, I., A remark on Rouché's theorem. Amer. Math. Monthly, 83, 1976, 186-187. Zbl0318.30002MR396919
  4. HURWITZ, A., Über die Nullstellen der Bessel'schen Funktion. Math. Ann., 33, 1889, 246-266. JFM20.0502.02
  5. LIPKA, S., Eine Verallgemeinerung des Rouchéschen Satzes. J. Reine Angew. Math., 160, 1929, 143-150. JFM55.0175.05
  6. LLOYD, N. G., Degree Theory. Cambridge Tracts in Math., 73, Cambridge University Press, 1978. Zbl0367.47001MR493564
  7. MARDEN, M., The Geometry of the Zeros. Math. Surveys, 3, Amer. Math. Soc., New York1949. Zbl0038.15303MR31114
  8. NACHBIN, L., Topology on Spaces of Holomorphic Mappings. Ergebnisse der Math, und ihrer Grenzgebiete, 47, Springer-Verlag, Berlin-Heidelberg-New York1969. Zbl0172.39902MR254579
  9. ROUCHÉ, E., Mémoire sur la série de Lagrange. J. Ecole Polytech., 22, 1862, 217-218. JFM01.0039.03
  10. SCHWARTZ, J. T., Compact analytic mappings of B-spaces and a theorem of Jane Cronin. Comm. Pure Appi. Math., 16, 1963, 253-260. Zbl0124.32703MR163178
  11. WLODARCZYK, K., Fixed points and invariant domains of expansive holomorphic maps in complex Banach spaces. Adv. Math., to appear. Zbl0868.47036MR1317617DOI10.1006/aima.1995.1010
  12. WHODARCZYK, K., The Rouché theorem for holomorphic maps in complex Banach spaces. Complex Variables Theory Appl., 20, 1992, 71-73. Zbl0789.46042MR1284353

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