No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Biholomorphic Mappings on Banach Spaces
Abstract
We present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.
ResearchGate has not been able to resolve any citations for this publication.
Zbl 0561.46022
Isidro, Jos ́e M.
;
Stach ́o, Laszl ́o L.
Holomorphic automorphism groups in Banach spaces: an elementary intro-
duction.
(English)
North-Holland Mathematics Studies, 105. Notas de Matem ́atica, 97. Amsterdam - New
York - Oxford: North-Holland. XII, 291 p. $ 44.50; Dfl. 120.00 (1985).
In the past 25 years or so, there has been an intensive development of holomorphy in
infinite dimensions. This is represented in a sense by the following books: 1)
L. Nach-
bin
, Topology on spaces of holomorphic mappings (1969; Zbl 0172.399); 2)
J. P. Ramis
,
Sous-ensembles analytiques d’une vari ́et ́e banachique complexe (1970; Zbl 0212.428); 3)
P. Noverraz
, Pseudo-convexit ́e, convexit ́e polynomiale et domaines d’holomorphie en
dimension infinie (1973; Zbl 0259.46049);
G. Coeur ́e
, Analytic functions and manifolds
in infinite dimensional spaces (1974; Zbl 0282.32015); 5)
T. Franzoni
and
E. Vesen-
tini
, Holomorphic maps and invariant distances (1980; Zbl 0447.46040); 6)
S. Dineen
,
Complex Analysis in locally convex spaces (1981; Zbl 0484.46044); 7)
J. F. Colombeau
,
Differential calculus and holomorphy (1982; Zbl 0506.46001); 8)
P. Mazet
, Analytic
sets in locally convex spaces (1984); 9)
H. Upmeier
, Symmetric Banach manifolds and
Jordan
C
∗
-algebras (1985; reviewed below). They are of a more or less advanced level.
The following ones are of an introductory nature: 10) the book under review; 11)
J. A.
Barroso
, Introduction to Holomorphy (1985); 12)
S. B. Chae
, Holomorphy and calculus
in normed spaces (1985); 13)
J. Mujica
, Complex analysis in Banach spaces (1985). One
of the important advances occured in connection with the problem of the determina-
tion of the holomorphic automorphisms of complex manifolds, to which the book under
review is primarily devoted. By combining the methods of the theories developed inde-
pendently by W. Kaup and J. P. Vigu ́e, the authors derive some of the main theorems
of this area.
The text is subdivided into Chapters 1 (uniformly bounded families of holomorphic maps
and locally uniform convergence), 2 (topological consequences of the group structure of
the set of automorphisms), 3 (the Carath ́eodory distance and completeness properties
of the group of automorphisms), 4 (the Lie algebra of complete vector fields), 5 (the
natural topology on the Lie algebra of complete vector fields), 6 (the Banach-Lie group
structure of the set of automorphisms), 7 (bounded circular domains), 8 (automorphisms
of the unit ball of some classical Banach spaces), 9 (bounded symmetric domains), and
10 (the Jordan theory of bounded symmetric domains).
Among the main features dealt with, we quote the following ones: holomorphic maps in
Banach spaces, Cartan’s uniqueness theorem, the Poincar ́e distance, the Carath ́eodory
pseudometric, Banach manifolds, Banach-Lie groups,
J
∗
-algebras of operators, and Jor-
dan triple product star algebras. The book ends with an extensive list of references and
supplementary reading adequate for its level.
L.Nachbin
This paper treats a holomorphic self-mapping f : Ω → Ω of a bounded domain Ω in a separable Hubert space ℋ with a fixed point p. In case the domain is convex, we prove an infinite-dimensional version of the Cartan-Carathéodory-Kaup-Wu Theorem. This is basically a rigidity result in the vein of the uniqueness part of the classical Schwarz lemma. The main technique, inspired by an old idea of H. Cartan, is iteration of the mapping f and its derivative. A normality result for holomorphic mappings in the compact-weak-open topology, due to Kim and Krantz, is used.
We describe bounded symmetric domains in complex Banach spaces and their biholomorphic automorphisms in terms of the underlying JB * - triple structures. Of particular interest in this context is the square root of a certain Bergman operator – we give a new description of this square root as exponential of a Hermitian operator which gives better norm estimates. We do not give full proofs. These will appear elsewhere.
In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces lp(Γ) and lp(Δ)(p > 1). We first derive the representation of isometries between the unit spheres of complex Banach spaces lp(Γ) and lp(Δ). Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces lp(Γ)and lp(Δ) can be extended to be a linear isometry on the whole space. © 2014 Wuhan Institute of Physics and Mathematics. Published by Elsevier Ltd. All rights reserved.
We present an infinite-dimensional version of Cartan-Carathéodory-Kaup-Wu theorem about the analyticity of the inverse of
a given analytic mapping. It is valid for a class of domains in separable Banach dual spaces that includes all bounded convex
domains.
Let ℋ be a separable, infinite dimensional, complex Hilbert space and let ℒ(ℋ) denote the algebra of all (bounded, linear) operators on ℋ. This paper is concerned with some aspects of the uniform approximation of an operator ℒ(ℋ) by nilpotent operators.