For large classes of complex Banach spaces (mainly operator spaces) we consider orbits of finite rank elements under the group of linear isometries. These are (in general) real-analytic submanifolds of infinite dimension but of finite CR-codimension. We compute the polynomial convex hull of such orbits explicitly and show as main result that every continuous CR-function on has a unique extension to the polynomial convex hull which is holomorphic in a certain sense. This generalizes to infinite dimensions results from a recent joint paper with D. Zaitsev in Inventiones math. 153, 45-104.
@article{ASNSP_2004_5_3_3_535_0,
author = {Kaup, Wilhelm},
title = {On the {CR-structure} of certain linear group orbits in infinite dimensions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {535--554},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {3},
year = {2004},
mrnumber = {2099248},
zbl = {1170.32314},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_3_535_0/}
}
TY - JOUR AU - Kaup, Wilhelm TI - On the CR-structure of certain linear group orbits in infinite dimensions JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2004 SP - 535 EP - 554 VL - 3 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2004_5_3_3_535_0/ LA - en ID - ASNSP_2004_5_3_3_535_0 ER -
%0 Journal Article %A Kaup, Wilhelm %T On the CR-structure of certain linear group orbits in infinite dimensions %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2004 %P 535-554 %V 3 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2004_5_3_3_535_0/ %G en %F ASNSP_2004_5_3_3_535_0
Kaup, Wilhelm. On the CR-structure of certain linear group orbits in infinite dimensions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 535-554. http://www.numdam.org/item/ASNSP_2004_5_3_3_535_0/
[1] - - , “Real Submanifolds in Complex Spaces and Their Mappings”, Princeton Math. Series 47, Princeton Univ. Press, 1998. | MR | Zbl
[2] - , A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), 387-421. | MR | Zbl
[3] , “CR Manifolds and the Tangential Cauchy-Riemann Complex”, Studies in Advanced Mathematics, CRC Press. Boca Raton, Ann Arbor, Boston, London 1991. | MR | Zbl
[4] , “Integration”, Hermann, Paris 1965.
[5] , “Complex Analysis on Infinite Dimensional Spaces”, Berlin-Heidelberg-New York, Springer, 1999. | MR | Zbl
[6] - , Enveloppes polynômiales d'ensembles compacts invariants, Math. Nachr. 266 (2004), 20-26. | MR | Zbl
[7] - , “Holomorphic Maps and Invariant Distances”, North Holland, Amsterdam, 1980. | MR | Zbl
[8] - , Linear algebraic groups in infinite dimensions, Illinois J. Math. 21 (1977), 666-674. | MR | Zbl
[9] , Algebraic Characterization of Symmetric Complex Banach Manifolds, Math. Ann. 228 (1977), 39-64. | MR | Zbl
[10] , A Riemann Mapping Theorem for Bounded Symmetric Domains in Complex Banach Spaces, Math. Z. 183 (1983), 503-529. | MR | Zbl
[11] , On spectral and singular values in JB-triples, Proc. Roy. Irish. Acad. 96A (1996), 95-103. | MR | Zbl
[12] , Bounded symmetric domains and polynomial convexity, Manuscripta Math. 114 (2004), 391-398. | MR | Zbl
[13] - , On the CR-structure of compact group orbits associated with bounded symmetric domains, Invent. Math. 153 (2003), 45-104. | MR | Zbl
[14] , “Jordan pairs”, Springer Lecture Notes 460, 1975. | MR | Zbl
[15] , Enveloppes polynomiales de compacts, Bull. Sci. Math. 116 (1992), 129-144. | MR | Zbl
[16] , “Randstrukturen beschränkter symmetrischer Gebiete”, Dissertation, Tübingen, 1995.
