Holomorphic Automorphism Groups in Banach Spaces

Name: Holomorphic Automorphism Groups in Banach Spaces | An Elementary Introduction
Brand: Elsevier
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An Elementary Introduction

J. M. Isidro L. L. Stachó(Autor*in)

Elsevier (Verlag)

Erschienen am 18. August 2011

290 Seiten

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Front Cover
HOLOMORPHIC AUTOMORPHISM GROUPS IN BANACH SPACES: AN ELEMENTARY INTRODUCTION
Copyright Page
TABLE OF CONTENTS
PREFACE
CHAPTER 0. PRELIMINARIES
CHAPTER 1. UNIFORMLY BOUNDED FAMILIES OF HOLOMORPHIC MAPS AND LOCALLY UNIFORM CONVERGENCE.
1. Cauchy majorizations.
2. Continuity of the composition operation.
3. Differentiability of the composition operation.
CHAPTER 2. TOPOLOGICAL CONSEQUENCES OF THE GROUP STRUCTURE OF THE SET OF AUTOMORPHISMS.
1. The topological group Aut D.
2. Cartan's uniqueness theorem.
3. Topological version of Cartan's uniqueness theorem.
CHAPTER 3. THE CARATHEODORY DISTANCE AND COMPLETENESS PROPERTIES OF THE GROUP OF AUTOMORPHISMS.
1. The Poincare distance.
2. The Carathéadory pseudometric.
3. The Carathéadory differential pseudometric.
4. Relations between the Carathéodory pseudometric and the norm metric on D.
5. Completeness properties of the group Aut D.
CHAPTER 4. THE LIE ALGEBRA OF COMPLETE VECTOR FIELDS.
1. One parameter subgroups.
2. Complete holomorphic vector fields.
3. The Lie algebra of complete holomorphic vector fields.
4. Some properties of commuting vector fields.
5. The adjoint mappings.
CHAPTER 5. THE NATURAL TOPOLOGY ON THE LIE ALGEBRA OF COMPLETE VECTOR FIELDS.
1. Cartan's uniqueness theorem for autD.
2. Some majorizations on autD.
3. The natural topology on autD.
4. autD as a Banach space.
5. autD as a Banach-Lie algebra.
CHAPTER 6. THE BANACH LIE GROUP STRUCTURE OF THE SET OF AUTOMORPHISMS.
1. The concept of a Banach manifold.
2. The concept of a Banach-Lie group.
3. Specific examples: the linear group and its algebraic subgroups.
4. Local behaviour of the exponential map at the origin.
5. The Banach-Lie group structure of AutD.
6. The action of AutD on the domain D.
CHAPTER 7. BOUNDED CIRCULAR DOMAINS.
1. The Lie algebra autD for circular domains.
2. The connected component of the identity in AutD.
3. Study of the orbit (AutD)o of the origin.
4. The decomposition AutD=(AutoD) (AutoD).
5. Holomorphic and isometric linear equivalence of Banach spaces.
6. The group of surjective linear isometries of a Banach space.
7 . Boundary behaviour and extension theorems.
CHAPTER 8. AUTOMORPHISMS OF THE UNIT BALL OF SOME CLASSICAL BANACH SPACES.
1. Some geometrical considerations.
2. Automorphisms of the unit ball of LP(O,µ),2?p?8.
3. Automorphisms of the unit ball of some algebras of continuous functions.
4. Operator valued Mijbius transformations.
4. Operator valued Möbius transformations.
5. J*-algebras of operators.
6. Minimal partial isometries in Cartan factors.
7. Description of Aut0B (F1) and aut0 B (F1).
8. Description of Aut0B (Fk) and aut0B(Fk).
CHAPTER 9. BOUNDED SYMMETRIC DOMAINS.
1. Historical sketch.
2. Elementary properties of symetric lomains.
3. The canonical decomposition of autD.
4. The complexified Lie algebra of autD.
5. The pseudorotations on autD.
7 . The pseudorotations on D.
8. The construction of the image domain D.
9. The isomorphism between the domains D and D.
CHAPTER 10. THE JORDAN THEORY OF BOUNDED SYMMETRIC DOMAINS.
1. Jordan triple product star algebras.
2. Polarization in J*-algebras.
3. Flat subsystems.
4. Subtriples generated by an element.
5. JB*-triples and Hermitian operators.
6. Function model for EC.
7. (Ec,*) as a commutative Jordan algebra.
8. positive J *- triples and the convexity of homogeneous circular domains.
9. Some properties of the topology of local uniform convergence.
LIST OF REFERENCES AND SUPPLEMENTARY READING

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