Holomorphic Automorphism Groups in Banach Spaces: An Elementary Introduction

H0LOM 0R PHIC AUTO M0R PHISM GROUPS IN BANACH SPACES: AN ELEMENTARY INTRODUCTION

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (97)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, and University of Rochester

NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

OXFORD

105

HOLOMORPHIC AUTOMORPHISM GROUPS IN BANACH SPACES: AN ELEMENTARY INTRODUCTION

Jose M. ISIDRO Facultad de Matematicas Universidad de Santiago de Compostela Spain

and LMO

L. STACHO

Bolyai lntezet Szeged Hungary

1985

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

@

Elsevier Science Publishers B.V., 1984

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87657 X

Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors forthe U.S.A. and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VAN DER BILT AVENUE NEW YORK, N.Y. 10017 U.S.A.

Library of Congress Cataloging in Publlcstion Data

Isidro, Joe6 M. Holomorphic automorphism groups in Benach spaces. (North-Holland nrsthematics studies ; 105) ( b a s de matendtics ; 97) Bibliography: p. 1. HolomrpMc functions. 2. Automorphism. 3. Banach spaces. I. StSch6, Lhsz16 L. 11. Title. 111. Series. Iv. s e r i e s : b t a s de m a t e d t i c a (Amsterdam, lctherlends) ; 97. W . n 8 6 no. 97 tQA333 510 s t515.9'83 84-21164 ISBN 0-444-87657-X (U.S. ) PRINTED IN THE NETHERLANDS

PREFACE

Since the early 70's,there has been intensive development in the theory of functions of an infinite number of complex variables. This has led to the establishment of completely new principles (e.g. concerning the behaviour of fixed points) and has thrown new light on some classical finite dimensional results such as the maximum principle, the Schwarz lemma and so on. Perhaps the most spectacular advances occurred in connection with the old problem of the determination of the holomorphic automorphisms of complex manifolds. This book is based on the introductory lectures on this latter field delivered at the University of Santiago de Compostela in October 1981 by the authors. Originally, it was planned as a comprehensive postgraduate course relying on a deep knowledge of holomorphy in topological vector spaces and infinik dimensional Lie groups. However, seeing that some of the undergraduate students were mainly interested in the study of bounded domains in Banach spaces, the authors restricted their attention to these aspects. This proved to be a fortunate idea. We realized that by combining the methods of the theories developed independently by W. Kaup and J . P . Vigu6 with minor modifications, even the main theorems could be derived. This was achieved in a self-contained way from the most fundamental principles of Banach spaces (such as the open mapping theorem), elementary function theory and the pure knowledge of the Taylor series representation of holomorphic maps in this setting. It may often happen in teaching mathematics that avodding the introduction of strong tools leads to abandoning natural heuristics. Probably, this is not the case now. It is enough to V

vi

PREFACE

recall how deeply the early development of the theory of finite dimensional Lie groups and Lie algebras was inspired in Cartan's investigation of the structure of symmetric domains. Moreover, we think that this approach to the automorphism groups of Banach space domains may also serve as motivating and illustrative material in introducing students to the theory of Lie groups and complex manifolds. The text is divided into eleven chapters. In chapter 0 we establish the terminology, and some typical examples of later importance (e.g. the Mtibius group) are studied. In chapter 1 we show the main topological consequences of the Cauchy estimates of Taylor coefficients for uniformly bounded families of holomorphic mappings. These considerations are continued in chapter 2 and applied specifically to the case of the automorphism group, concluding with the topological version of Cartan's uniqueness theorem. The global topological investigations finish in chpater 3, where the Caratheodory distance is introduced to obtain the completness properties of the group AutD. In chapter 4 a completely elementary introduction to Lie theory begins by showing where one-parameter subgroups come from. Chapter 5 is devoted to a description of the Banach Lie algebra structure of complete holomorphic vector fields in order to lay the foundation of chpater 6, in which the Banach Lie groups structure of AutD is studied. In chpaters 7 and 8 we discuss the basic theory of circular domains and determine explicitly the holomorphic automorphism group of the unit ball of several classical Banach spaces. In chapter 9 we introduce the reader to another fruitfully developing branch of these researches by proving Vigue's theorem on the Harish-Chandra realization of bounded symmetric domains. Finally, in chapter 10 and elementary introduction of the Jordan approach to bounded symmetric domains is presented and the convexity of the Harish-Chandra realization is proved. We would like to express our sincere acknowledgement to Prof. L. Nachbin who suggested the idea of writing these notes

PREFACE

vii

and who, together with Prof. E. Vesentini, introduced the authors to infinite dimensional holomorphy and this fascinating branch of mathematics. Thanks are also due to M. Teresa Iglesias €or the careful typing. The authors, August 1984.

J.M. Isidro

Santiago de Compostela Spain.

Stach6 Szeged Hungary.

L.L.

This Page Intentionaiiy Left Blank

TABLE OF CONTENTS

PREFACE

V

CHAPTER

0. PRELIMINARIES.

CHAPTER

1 . UNIFORMLY BOUNDED FAMILIES OF HOLOMORPHIC

§l.

52. 13.

CHAPTER

51. 52. 53.

CHAPTER

51. 52.

53. 54. 55.

1

MAPS AND LOCALLY UNIFORM CONVERGENCE. Cauchy majorizations. Continuity of the composition operation. Differentiability of the composition operation.

5

9 13

2 . TOPOLOGICAL CONSEQUENCES OF THE GROUP

STRUCTURE OF THE SET OF AUTOMORPHISMS. The topological group Aut D Cartan's uniqueness theorem. Topological version of Cartan's uniqueness theorem.

.

17 19 20

3. THE CARATHEODORY DISTANCE AND COMPLETENESS

PROPERTIES OF THE GROUP OF AUTOMORPHISMS. The Poincar6 distance. The Caratheadory pseudometric. The Caratheadory differential pseudometric. Relations between the Carathgodory pseudometric and the norm metric on D.

32

Completeness properties of the group Aut D.

37

ix

29

33 35

TABLE OF CONTENTS

X

CHAPTER 51. 52. 53. 54. 55.

CHAPTER 51.

52.

53. 54. 55.

CHAPTER 51. 52. 53.

54.

55. 56. CHAPTER §I.

52.

53. 54. 55.

4. THE LIE ALGEBRA OF COMPLETE VECTOR FIELDS. One parameter subgroups. Complete holomorphic vector fields. The Lie algebra of complete holomorphic vector fields. Some properties of commuting vector fields. The adjoint mappings. 5. THE NATURAL TOPOLOGY ON THE LIE ALGEBRA OF COMPLETE VECTOR FIELDS. Cartan's uniqueness theorem for autD. Some majorizations on autD. The natural topology on autD. autD as a Banach space. autD as a Banach-Lie algebra.

6. THE BANACH LIE GROUP STRUCTURE OF THE SET OF AUTOMORPHISMS. The concept of a Banach manifold. The concept of a Banach-Lie group. Specific examples: the linear group and its algebraic subgroups. Local behaviour of the exponential map at the origin. The Banach-Lie group structure of AutD. The action of AutD on the domain D. 7. BOUNDED CIRCULAR DOMAINS. The Lie algebra autD f o r circular domains. The connected component of the identity in AutD. Study of the orbit (AutDIO of the origin. 0 The decomposition AutD=(Aut D)(AutoD). Holomorphic and isometric linear equivalence of Banach spaces.

43

49 54

58

61

65 66

69 70

74

77 83 87 101 108 111

113

120 124 126 128

TABLE OF CONTENTS 56. 57.

CHAPTER 51.

52. 53. 54.

55. 56. 57. S8.

CHAPTER §1.

52. 53. 54. 55. 56.

§7.

58. 59.

CHAPTER 51. 92. 53.

54 *

55.

96. 57.

xi

The group of surjective linear isometries of a Banach space. Boundary behaviour and extension theorems.

130 132

a.

AUTOMORPHISMS OF THE UNIT BALL OF SOME CLASSICAL BANACH SPACES. Some geometrical considerations. Automorphisms of the unit ball of LP(Q,p),2#p#m. Automorphisms of the unit ball of some algebras of continuous functions. Operator valued Mijbius transformations. J*-algebras of operators. Minimal partial isometries in Cartan factors. 0

0

Description of Aut B(F1) and aut B(F1). Description of Aut 0 B ( F k ) and aut0B(Pk).

139 142 148 157 164 169

178 183

9. BOUNDED SYMMETRIC DOMAINS.

Historical sketch. Elementary properties 3f svmetric lomains. The canonical decomposition of autD. The complexified Lie algebra of autD. The local representation of autD. The pseudorotations on autD. The pseudorotations on D. The construction of the image domain 8. The isomorphism between the domains D and 8.

191 193 199 20 1 203 207 213 221 224

10. THE JORDAN THEORY OF BOUNDED SYMMETRIC

DOMAINS. Jordan triple product star algebras. Polarization in J*-algebras. Flat subsystems. Subtriples generated by an element. JB*-triples and Hermitian operators. Function model f o r EC. (Ec,*) as a commutative Jordan algebra.

231 235 238

240 242 249 262

x ii

TABLE OF CONTENTS

98.

p o s i t i v e J * - t r i p l e s and t h e c o n v e x i t y of homogeneous c i r c u l a r d o m a i n s .

270

59.

Some p r o p e r t i e s of t h e t o p o l o g y of l o c a l uniform convergence.

280

L I S T OF REFERENCES AND SUPPLEMENTARY READING

285

CHAPTER

0

PRELIMINARIES

Throughout what follows, E and E l denote complex Banach spaces whose norms will be represented indistinctly by ]I I ] , and D is a bounded domain in E.

-

0.1. DEFINITION. A m a p p i n g f: D+E1 i s s a i d t o b e h o l o m o r p h i c i f , for e v e r y aeD, we h a v e f

(a+h)=

. . - ,h)

m

C fLn (h, n=O

i n a n e i g h b o u r h o o d o f a.

Here, for every ndN,

a"

1

(0.1)

f(a+tlhl+. . . + t h 1 n n

is a continuous n-linear operator from En into E l . Remark that, f o r ndN and hcE, we have (0.2)

The family of all holomorphic mappings from D c E into a set D 1 C E 1 is denoted by Hol(D,D1). When E = E 1 and 9=D1 we write H o l ( D ) instead of Hol(D,D ). 1

0.2. DEFINITION. A s u b s e t B c D i s s a i d t o b e c o m p l e t e l y i n t e r i o r t o D, a n d we w r i t e B C C D, if dist(B,aD) > O .

For feHol(D,D1) and B c c D we define

I / f 11

I/ f /I

B=: SUP xeB

IIf

by means of (XI

I1

0.3. DEFINITION. A n e t ( fI. ) , i n Hol(D,D 1 is s a i d t o JCJ c o n v e r g e locally u n i f o r m l y t o a m a p p i n g fcHol(D,E1) i f , f o r 1

CHAPTER

2

0

We denote by 7 the topology on Hol(D,D1) of local uniform convergence over D. If a net (f , ) , in Hol(D,D ) is locally 3

JCJ

1

uniformly convergent to fcHol(D,El), we write T= lim f . = f jeJ

'

0.4. EXERCISE. (a) Let E be the Banach space !Z1 and

function f: E-E is holornorphic on the whole space E and that f is not bounded on the open unit ball B ( E ) of E. Thus we may have 1 1 f / l B =m even if fcHol(D,El) and B c c D. (b) Is T a metrizable topology?. 0.5. DEFINITION. A m a p p i n g fcHol(D) is s a i d to b e an nutomorphism

oS D

if t h e r e exists gcHol(D) s u c h t h a t fg= id

n = gf

Here fg stands f o r the composite of the mappings f and g, and idD represents the identity mapping of D. The family of all automorphisms of D is represented by AutD. 0.6. EXERCISE, (a) Prove that a mapping f: D+E satisfies fcAutD if, and oniy if, f is a surjective bijection of D and, for every asD, the operator ':f is invertible.

invertibility of f

(b) Can the assumption concerning the be weakened?.

(c) Show that AutD with the usual law of composition is a group.

0.7. EXAMPLE. Let A be the open unit disc of E and, for k,ucQ with \ k l = 1 and l u l < l , let us define M as the restrick,u tion to A of the Mdbius transformation

PREL IMI N A R I €3S

3

Then, t h e f o l l o w i n g r e s u l t h o l d s : 0.8.

The g r o u p AutA is g i v e n b y

THEOREX.

P r o o f : F i r s t , l e t us o b s e r v e t h a t w e have

where

1w1 1

I

. After

of t h e argument w e g e t f , + f r e l a i i v e t o I

f j

aicBL-l f o r

k

serveral reiterations

I( - I I B ,

n+ 1

f.+f r e l a t i v e t o 3

. Thus

I / . 11

B2

.

B1 and B2 may be changed, t h e proof i s c o m p l e t e .

#

BOUNDED

FAMILIES AND UNIFORll CONVERGENCE

9

1 . 7 . COROLLARY. T h e t o p o l o g y T o n Hol(D,D ) i s m e t r i z a 1 is a m e t r i c o n Hol(D,D1) //

b l e . F o r any baZZ B c c D ,

We h a v e Tlimf.= f

i n Hol(D,D1) 3 f!k +f(k f o r a l l ],a a kcPI, o r if and o n l y if t h e r e e x i s t s a b a l l B c c D s u c h t h a t

whose a s s o c i a t e d t o p o Z o g y i s 2'.

i f , and onZy if, t h e r e e x i s t s aeD s u c h t h a t

/ I fj-fll, 52.-

+o-

Continuity of the composition operation.

Let D,D1 and D2 be bounded domains in the Banach spaces E, and E 2 . As a first application of the previous theorem we show that the composition of mappings Hol (D,D1)XHol (D1,D2)+Hol (D,D2)

is continuous with regard to the topology of local uniform convergence. The way we shall follow is perhaps not the shortest possible but it provides information that turns out to be useful later. 1.8.

PROPOSITION. L e t feHol(D,D1) and qeHol(D1,D2) b e

h o Z o m o r p h i c m a p p i n g s whose r e s p e c t i v e T a y l o r ' s s e r i e s a t acD and b=: f (a)eD a r e

f(a+h)= f(a)+

...,h)

C fin(h, n=1

f o r a l l kCN.

Here, the detailed interpretation of ( 1 . 4 )

10

CBAPTER

v

1

1

+...+ vm>1

Pnoof: We have the following formal expansion for gf about the point asD (1.5)

gf (a+h)= g(b) +

?

m= 1

.

rjLm[f (a+h)-b;. .;f (a+h)-b]=

We point out that (1.5) is uniformly convergent in a suitable neiqhbourhood of a. Indeed, by ( 1 . 1 ) we have the majorizations

13 where 6=: dist(a,aD) and

E=:

dist(b,aD1

.

v +..+v

Hence

m

BOUNDED FAMILIES AND UlJIFORl4 CONVERGENCE

11

whenever

which i s s a t i s f i e d f o r s u f f i c i e n t l y s m a l l v a l u e s of f o r 1 1 hlI j o . Thus g T 1 + i d D w i t h r e g a r d t o 1

11

- /IB

and t h e result

#

f o l l o w s from t h e o r e m 1 . 6 . 2 . 3 . EXERCISE. Show by e x a m p l e s i n Aut A t h a t t h e c o n s t a n t K i n lemma 2 . 1 , B'

.

2.4.

i n g e n e r a l , must d e p e n d o n b o t h B a n d

REImRK. F o r a b e t t e r u n d e r s t a n d i n g of t h e s i t u a -

t i o n i n lemma 2 . 1

it i s i m p o r t a n t t o n o t e t h a t :

Given a n y b a l l B c c D , t h e r e i s a c o n s t a n t K s u c h t h a t w e have

19

TOPOLOGY ON THE GROUP OF AUTOMORPHISMS

f o r a l l f,gcAut D. Proof: L e t B ' = B

s/2

be d e f i n e d as above,where a g a i n

6=: d i s t ( B , a D ) , a n d c o n s i d e r a n y xsB. NOW, i f t h e p o i n t y = : g - l f ( x ) l i e s i n B ' , t h e n by p r o p o s i t i o n 1 . 4 w e h a v e

T h u s , by t h e a r b t r a r y n e s s o f xsB w e o b t a i n

I/ 2.5.

f-9

EXERCISE. U s i n g t h e f a c t t h a t c o n s t a n t m a p p i n g s

h a v e n u l l d e r i v a t i v e , show

f o r a l l f , gsAut D.

52.-

t h a t w e have

Hint: s h i f t D.

C a r t a n ' s uniqueness theorem. _____-

Next w e i n v e s t i g a t e t h e c o n s e q u e n c e s o f t h e € a c t t h a t , i n Aut D ,

t h e c o m p o s i t i o n c a n be i n f i n i t e l y i t e r a t e d . The p r o o f o f

t h e r e s u l t w e s h a l l o b t a i n s h a d s t h e f i r s t l i g h t on how t h e g e o m e t r y of D d e t e r m i n e s t h e a u t o m o r p h i s m s .

f:'=

9:'

2.6.

THEOREM.

L e t f , geAut D . I f for some acD

and

f ( ' = 9:'

then f = q.

we h a v e

P r o o f : L e t u s c o n s i d e r t h e map h = q - l f . W e h a v e h ( ' = (1

and h a = i d . T h e r e f o r e it s u f f i c e s t o p r o v e t h e s t a t e m e n t o f t h e theorem f o r h=: i d D . Suppose h f i d , .

Then t h e r e e x i s t s some R d N s u c h t h a t

a

CHAPTER

20

h',

C o n s i d e r t h e i t e r a t e d maps h ' ,

2

...

W e show by i n d u c t i o n t h a t

f o r a l l p a . Obviously a

( h p ) ('=

and

(hp)il= id

f o r a l l p m , and t h e a s s e r t i o n of

( 2 . 3 ) f o r p= 1 i s nothing but

t h e d e f i n i t i o n of k . Assume ( 2 . 3 ) h o l d s f o r p. By p r o p o s i t i o n 1 . 8 w e have

vl,.

C o n s i d e r t h e c a s e 2 g k i R . Then, from

f o r k = 1,2,...

(hp)ik=0

( h p ) ( 2 = '.-.=

. , v m 31

and

L e t u s compute ( h P t l ) i ' .

( h P f l ) ('=

(hp)

hJk= 0

w e d e r i v e ( hP + l ) (a= 0

From ( h p ) i 2 = . . = (hP);'-'=

. . , h a( 1] =

[hi'] + ( h p ):'[h;',

hi'cp

0

we get

hi'=

(p+l)h,('

which p r o v e s ( 1 . 3 ) . Hence w e have

lirn

11

(hp)i'II =

m

which c o n t r a d i c t s t h e Cauchy

P*m

majorizations

//

(Chapter 1 , p r o p o s i t i o n 1 . 1 ) ( h p ) ;'I\

6 (

e

)

2

sup(

11 X I \

i

xsD1

f o r a l l p a . I n f a c t w e have p r o v e d t h e f o l l o w i n g : 2.7.

COROLLARY.

L e t hsHol(D) be g i v e n and a s s u m e t h a t (1

t h e r e e x i s t s a p o i n t acD for w h i c h we h a v e h ( O = a , ha = i d . Then h= i d

D

.

e n e s s theorem. Roughly s p e a k i n g , C a r t a n ' s u n i q u e n e s s t h e o r e m s t a t e s t h a t ,

TOPOLOGY ON THE GROUP OF AUTOMORPHISMS

21

g i v e n a p o i n t a e D , t h e a u t o m o r p h i s m s of D d e p e n d o n l y on t h e i r 0 - t h a n d I-st d e r i v a t i v e s a t t h e p o i n t a , i . e . , a n y f s A u t D i s u n i q u e l y d e t e r m i n e d by t h e p a i r f (O a

t

f

('.

d

Is t h i s correspon-

d e n c e c o n t i n u o u s ? . The a n s w e r i s a f f i r m a t i v e . W e h a v e t h e f o l l o w i n g t o p o l o g i c a l v e r s i o n of C a r t a n ' s uniqueness theorem: 2.8.

THEOREM.

L e t asD b e f i x e d a n d a s s u m e t h a t

f , f . c A u t D, jsJ, s a t i s f y I Tlim f = f. I jeJ

,

f o r s= 0 ,1 . T h e n we h a v e

f ! ",a+f:s

P r o o f : E s s e n t i a l l y , w e c a r r y o u t t h e r e a s o n i n g s of t h e p r o o f o f C a r t a n ' s t h e o r e m i n a more g e n e r a l s e t t i n g ( w h e r e t h e r o l e o f f i s now p l a y e d b y f , ) . I D e f i n e h . = : f - ' f . . The r e l a t i o n h ! * = f - ' [ f j ( a ) ] + a 1 I I Observe t h a t also

is clear.

i n a c c o u n t of p r o p o s i t i o n 1 . 1 . N e x t w e s t a b l i s h t h e f o l l o w i n g a x u l i a r y stament LEMMA. I f

2.9.

h i o +a Ira

,

h!'

+id

1 la

and

i s a n e t in H o l ( D ) s u c h t h a t

(h,),

(kl IeJ

h . +O Ira

for k = 2 , .

.., R -I

t h e n we h a v e

An i n m e d i a t e c o n s e q u e n c e w e o b t a i n 2.10.

COROLLARY. F o r a n y n e t

( h . ) , c H o l ( D ) with I IeJ

I n d e e d : I f w e h a d h j f t i d D t h e n by t h e o r e m 1 . 5 t h e r e would be a n R > 2 s u c h t h a t h(.k

-to

3 ,a

Then, b y lemma 2 . 9

f o r 26kSR

and

22

CHAPTER

2

for all j c J and pm. Therefore

-

l i m //(hy);'l/

bpX

jeJ

for pm. But this contradicts the Cauchy estimates (proposition 1.1).

p r o v i n g the corollary. T h u s , in our case f-l f , - + idD whence Tlim f , = f. j

1

e

'

~

Thus, o u r only remainder task i s to prove the lemma. This requires a better overlook on the expansion of h p as a direct iteration of the formula given by proposition 1.8 would lead to very involved expressions. Instead, let us procceed a s follows: Start from

This a s s e r t s that, for f , gsHol(D) a n d for sufficiently small vectors xeE, (gf)(a+x) is the sum of all possible expressions

Let us write (2.4) i n the more visualizable form

/I\

x..x..x

...

/I\

x..x..x

...

/I\

x..x..x

TOPOLOGY ON THE GROUP O F AUTOMORPHISMS

23

I n s u c h a way, it seems t o b e i n t u i t i v e l y c l e a r t h a t f P ( a + x )

i s t h e sum of a l l p o s s i b l e e x p r e s s i o n s c o r r e s o n d i n g t o t h e g r a p h s of t h e form

.....

x

x

x

x

H e r e t h e symbol

7(

.....

.....

x

x

.....

x

x

x

x

x

x

c a n b e i n t e r p r e t e d a s t h e s i g n of s u b s t i t u -

t i o n . Now w e s t a b l i s h t h e p r e c i s e mathematical development of t h i s technique.

I n o r d e r t o be s e l f - c o n t a i n e d ,

w e s h a l l make

no r e f e r e n c e t o t h e u s u a l t h e o r y of t r e e g r a p h s .

2.11.

DEFINITION. L e t nCN b e a r b i t r a r i l y S i z e d . A n-up2.e A =

" t r e e of h e i g h t n r r i s a n

(ao,..,a

n- 1

1 of f u n c t i o n s

such t h a t t h e d o m a i n of a

11 f o r e v e r y p , dom

P

i s a segment

=tl,2,..dp(A)loflN. P r a n g e a,-1= { I } .

c1

31

" o n t o " dom

for Ogpgn-1 LX

Pf 1

I

.

41 for p = I , . . , n

a

P

i s a "monotone i n c r e a s i n g " mapping

we h a v e dom

c1P =

range ap - 1 *

The number d ( A ) i s c a l l e d t h e w i d t h o f A a t t h e h e i g h t p , and W e s a y t h a t d ( A ) i s t h e d e g r e e of

we s h a l l w r i t e d(A)=: d 0 ( A ) . A.

24

CHAPTER 2

2.12.

Note

C o n s i d e r t h e p l a i n graph

EXAMPLE.

t h a t t h e s e q u e n c e of t h e v e r t i c e s i s r e l e v a n t ! This c a n

b e i n t e r p r e t e d a s a t r e - of h e i g h t 3 a s follows:

0

1

a 1 (2)= 1

(I)= 1

a o ( l ) =1

... cx 0 ( 5 ) =

3

,.,

a1(5)= 2

...

~1

0

(10)=5

A t t h i s p o i n t w e c a n p r o v i d e a n e x a c t i n t e r p r e t a t i o n of

2.13.

(2.5).

L e t T r e e s ( n ) d e n o t e Ihe s e t o f a l l

DEFINITION.

t r e e : : o f h e i g h t n. G i v e n g e H o l ( D ) ue d e f i n e t h e " t r e e - d e r i v n t i ves

g i A of g at a c D a s foZZows:

F o r AcTrees ( 1 ) w e s e t g

(A

=:

(d(A) i n t h e u s u a Z s e n s e . ga

giH is a l r e a d y d e f i n e d for a l l BcTrees (n-1) and A=

( C X ~ ~ . ).€ ~ T rC e eX s n- 1

( n ) we s e t

If

TOPOLOGY ON THE GROUP OF AUTOMORPHISMS

25

-1

)sTrees(n-l), a 0 (k)=: {m: ao(m)= k } a n d Jf means cardinality. ljhere B= (al,..,cx

n- 1

2.14. PROPOSITION. T h e series 1 AeTrees (n)

I1

(A

4,

(xr ..rxI

I1

is uniformly convergente o n some neighbourhood o f the origin and we have

Proof: For n= 1 this formula is equivalent to the usual Taylor expansion of g. Remark that by the Cauchy estimates we have

where 6 = dist(a,aD)

and p ~ [ O , m ) .

NOW, assume we had proved (2.71

gn-' (b+x)= g"-' (b)+

C

BeTrees ( n )

gLE(x,* * r x )

for sufficiently small vectors x and

for all

pc[Orm),

where W

w

Y

(PI=

ep

1 [ v = 1 dist (y,aD)

1"

We prove (2.7) and (2.8) for n. Let us begin with (2.8)

CHAPTER

26

2

there exists a But given any BcTrees(n-I) and v l I . . , v d(B) unique (Y. such that (aolR)cTrees(n) and w k = ## a 0- 1 ( k ) f o r 0 k = l,..ld(B). Thus the second member of (2.9) is

which, due to Cauchy's majorizations, is dominated by m

c

c RcTrees(n-1)

V

1

+..tV

d

(R)

By the induction hypothesis the last sum is dominated by

Thus (2.8) is established. Now (2.8) is immediate

c g;* ( X I ,.x)= AeTress (n)

= g n (a+x)-gn

.

(a)

#

27

TOPOLOGY ON THE GROUP OF AUTOMORPHISM

Now w e c a n p r o v e t h e lemma. Proof o f lemma 2 . 9 :

I t f o l l o w s from p r o p o s i t i o n 2 . 1 4

f o r any p,kcW, acD and qcHol(D)

.

that

Consider any AsTrees(p)

with

d ( A ) = R s u c h t h a t A h a s a v e r t e x where t h e number o f e n t e r i n g e d g e s i s d i f f e r e n t from 1 and R , A=

)

( ~ 1 ~ , . . , ~ 1

that is, i f

t h e n t h e r e a r e s , udN w i t h

P-1 # c ~ - ~ ( u ) f ?W . e show t h a t , i n t h a t case, !h

# c ~ ~ ~ ( v and ) f l

+O. 3 ,a

Indeed, it i s

e a s y t o see t h a t

t h a t i s , 11 h! ,(fI/ i s n o t g r e a t e r t h a n t h e p r o d u c t of t h e norms of a l l t h o s e d e r i v a t i v e s t h a t o c c u r a t some v e r t e x o f A i f w e draw h ( A as j,a

I 1

h ( * . . . h a( *

...

I I (*

... h ( * . ..ha

By Cauchy m a j o r i z a t i o n s a l l t h e f a c t o r s o f

(2.10)

are bounded

by a c o n s t a n t i n d e p e n d e n t o f j . A t t h e same t i m e , by c o r o l l a r y 1 . I 1 we have

CHAPTER

28

2

and k f l e n t a i l s h ! k +O = i d ( k a n d h e n c e 3 ,a b e c a u s e h s ( a ) * a . The o n l y t r e e s A i n T r e e s ( p ) w i t h

since k= # a - ' ( v ) < L

-to

h(k

j ,hs ( a )

I t h e p r o p e r t i e s t h a t e a c h of t h e i r v e r t i c e s a d m i t a 1 o r Q e n t e r i n g edge and d ( A ) = L a r e

... ... ... ... ... I

I

I

. . . .. .. .. ..

p

I

1

L e t u s c a l l them AlI...,A

h. (A"

(e

,,a *ha

P

for

2 . 1 5 . EXERCISE. i n t h e g r a p h form.

1

I P

1

R

R

Therefore

/

p . . .

. Observe

V=

1I

.

that

. p , whence

( a ) Write t h e series of f n f n - 1 - .f 1 (b) Prove ( 2 . 1 0 ) .

CHAPTER

3

THE CARATHEODORY DISTANCE AND COMPLETENESS PROPERTIES OF THE GROUP OF AUTOMORPHISMS

51.-

The Poincar6 distance.

3.1. DEFINITION. We s a y t h a t a m e t r i c d o n a b o u n d e d d o m a i n D i s Aut D - i n v a r i a n t i f w e h a v e

f o r a l l x , ysD a n d a l l fcAut D . 3 . 2 . LE+W.The f u n c t i o n

d e f i n e s a n Aut A - i n v a r i a n t

m e t r i c o n A.

As usually, we write tanh(dAl$

I/

x-YII

q ) ( X l Y ) $

p1 II

IK

'

or m o r e

x-YII

f o r a l l x,ycK.

P r o o f : Given yeX, l e t B d e n o t e t h e b a l l of c e n t e r y and r a d i u s diam(D)

.

S i n c e B T D , w e h a v e d B I D s d D .But

d B ( x , y )= t a n h

f o r a l l xsB b e c a u s e t h e f u n c t i o n t a n h - I .is c o n v e x on LO,=)

i t s d e r i v a t i v e a t t h e p o i n t 0 i s equal t o 1 , h e n c e t a n h for a l l t>O.

This proves t h e left-hand

1

and F,>
some aeD we h a v e f , ( a ) + b s D . T h e n THEOREM. L e t

3.15.

there exists

fsAut D

such t h a t T l i m f . = f . I

I

JCJ

P r o o f : C l e a r l y w e h a v e a u n i q u e mapping f e H o l ( D , E ) w i t h Tlim f = f . T h u s , i f were f e A u t D , by t h e o r e m 2 . 2 we would 3 IeJ h a v e T l i m f T 1 = f-l. jeJ

1

We show t h a t -1

( f j )jcJ

(3.7)

i s a T-Cauchy

sequence.

L e t u s w r i t e b . = : f , ( a ) and c h o o s e a n y b a l l s 1

3

B c c D and

B ' c c D c e n t e r e d r e s p e c t i v e l y a t a and b s u c h t h a t f ( B ) c B ' .

From t h e o r e m 3 . 1 3 i t f o l l o w s t h a t

Hence w e c a n f i x 6>0 s o t h a t t h e set C=

i s contained i n B. L e t u s i n t r o d u c e a l s o Cf=: 1 W e may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t d D ( b . , b ) 0 with B g c c D ) . Then

for all xeB and

te[-toIt0]. Therefore

56

4

CHAPTER

f o r all xeB and t , Itl- O w i t h B g C C D a n d that c ~ e h a v e

gl(

,

<m

t h e r e is n n u m b e r r > O such

Bfi t

t k

m

Ftg(x)=:g[f ( x ) ] =

I;

*

(A g ) (x)

k=O

I t l s i . T h e s e r i e s c o n v e r g e : : uniformly o n

w h e n e v e r x8B and

1 -

P r o o f : For t > O w e s e t A =:

. Therefore

Tlim A = A t

t+O

4.17.

(fT-id 1. W e have

t

t h e s t a t e m e n t i s a consequence of L e t A l l A2eAut D b e g i v e n .

THEOREM.

B.

(4.4).

Then A1+A2

a n d A ( ' A ~ - A ; ~ A2 b e Z o n g t o a u t D . 1

= g[exp(tA

)]

(1

=: g+Ag g and F . = : g+qft= 1 where g s H o l ( D , E ) , te2R and j = 1 , 2 . C o n s i d e r t h e

Proof: L e t u s w r i t e

n e t ( gt ) c A u t D d e f i n e d by gt=: f : Tlim(ft-id,)= t+O

A +A 1

2'

A

t

g

f:.

W e show t h a t

I n d e e d , g i v e n any b a l l B E D ,

for

s u f f i c i e n t l y small v a l u e s o f t w e have 'tlB

=

gt2

?:(idg)=

[exp t

*

i22][exp

t A l l ( i d B )=

*

= i d + t ( A Z + A 1 ) ( i d B )+. .= i d + ( A + A 2 ) B B 1

Therefore

t1

(gt-id )

IB

+A +A2 i n 1

(1 - I /

+.

.

B. By t h e a r b i t r a r i n e s s

of B , t h e convergence h o l d s a l s o i n t h e t o p o l o g y T. Then theorem 4 . 5 e s t a b l i s h e s t h a t A +A2&aut D . 1

The proof of A i 1 A 2 - A 2 ( 1A s a u t D i s s i m i l a r 1

by c o n s i d e r i n g t h e

COMPLETE VECTOR FIELDS

n e t h =: t

fi

f:

1

T l i m - (h -id t+O t 2

fit )=

57

fYt and showing t h a t (1

A(1 A2-A2 1

4.18. DEFINITION. G i v e n A1 ,A2eHol(D,E), we d e f i n e (1

[A,,A,I=: A:~A2 - A ~A 1 T h e r e f o r e w e have

for xsD. The o p e r a t i o n A2

c,]

i s c a l l e d t h e L i e p r o d u c t of A1 and

*

For f i x e d AsHol(D,E) , t h e l i n e a r o p e r a t o r [A, .] i s c a l l e d t h e

a d j o i n t of A and w i l l be denoted by A

W '

4.19. PROPOSITION. F o r e v e r y AcHo~(D,E), t h e adjoint of A is d e r i v a t i o n o n a u t D, i . e . , we h a v e

f o r a l l A1,A2cHol(D,E). Proof: Since t h e Lie product i s c l e a r l y anticommutati-

ve, a l l w e have t o prove i s t h e J a c o b 3 i d e n t i t y

f o r a l l A1,A2,A3eHol(D,E). B u t

= A

(1 (1

1

(1 (1

(1

(1

A 2 A3-A1 A 3 A2-(A2 A3-AJ AZ) ('A 1 =

-

Summing up t h e s i m i l a r e x p r e s s i o n s f o r t h e c y c l i c p-errnutations of t h e i n d e x e s w e o b t a i n t h e d e s i r e d r e s u l t . iy

CHAPTER

58

4

4.20. DEFINITION. An a l g e b r a U i t h a p r o d u c t

[,I

is

caZZed a " L i e a l g e b r a " if [,] is a n t i c o m m u t a t i v e a n d sntisfies the Jacvbi i d e n t i t y .

Thus we have proved 4.21. THEOREM. T h e s e t aut D i s a r e a l L,Le a l g e b r a w i t h r e s p e c t t o t h e p r o d u c t IAIIA2]=: A 1( 1 A2-A2( 1A 1 . dk

4.22. EXERCISES. ( a ) Show that dtk

la

(exptAl)..(exptAn)

belongs to the Lie subalgebra of aut D generated by A1,..A,. (b) Prove that we can write

T=

-' in corollary 4 . 1 3 .

811 All

B6

54.- Some properties of commuting vector .fields. __- Now we turn to the investigation of holomorphic vector fields A,BsHol(D,E) with the property %= ii. In general,

i6X=

i ( X ( l B ) = (X ( ' B ) ('A=

2x( 2( B , A ) +x('B('A

and hence

that is,

..

[A,B] = BA-AB

Thus,

and

6

commute if and only if [A,B]= 0 .

Furthermore, we remark that if XcHol(D,E) is an arbitrary vector field then, using the argument leading to (4.7) we get (4.16)

for any open ball B with B 6 c C D , any faHol(D,E) and n m . Therefore we have (exptX)x=

-c

n=O

t" 3

(Xnid,)xeBg

COMPLETE VECTOR FIELDS

6

59

-1

I t \ O s u c h t h a t B f i c C D a n d

k

j o . Then, by ( 5 . 4 ) w e have d

( a ) e B ' = : B,,2 f o r a l l j > j o and k = 1 ,

*

*

lPj

(a)

. Therefore,

by p r o p o s i t i o n 1 . 4

and ( 5 . 2 )

J

whenever j > j o and 1CkCp.. Taking i n t o a c c o u n t ( 5 . 4 ) and t h e I f a c t t h a t /I f,-idDII 0,we can w r i t e 3

B

-+

f o r some c o n s t a n t y ( i n d e p e n d e n t of j , k ) and a l l j ? j k= l , . . , p j .

0'

Hence,

(5.5) f o r ] > l o . I t i s w e l l known from e l e m e n t a r y a n a l y s i s t h a t

R

( l + a . ) I +I

I

whenever a

j

+

0 and

c1

j

P

-+

j

0 . B u t , by ( 5 . 3 1 ,

69

TOPOLOGY ON VECTOR F I E L D S

5.4. B’cCD

B,

COROLLARY.

F o r e v e r y p a i r of ba2l.s B and B ’ w i t h

t h e r e e z i s t s a c o n s t a n t K ’ s u c h t h a t we h a v e

for all f , gsAutD s a t i s f y i n g f ( B ) C B ’ and g ( B ) C B ’ . P r o o f : Given f , gsAutD w i t h f ( B ) c B ’ a n d g ( B ) C B I ,

by

t h e o r e m 5 . 3 a n d remark 2 . 4 w e h a v e

4K2

4CK

2

+K2

w h e r e , by p r o p o s i t i o n 1 . 4 ,

t h e right-hand

s i d e i s d o m i n a t e d by

f o r some c o n s t a n t s K 1 ’ K 2 , K 3 d e p e n d i n g o n l y on B a n d B ’ .

ff

5 . 5 . E X E R C I S E . L o o k f o r c o u n t e r e x a m p l e s t o show t h a t t h e c o n s t a n t K ’ i n c o r o l l a r y 5 . 4 must a c t u a l l y d e p e n d o n B a n d B’

.

53.-

The ~- n a t u r a l t o p o l o g y on a u t g .

From t h e s t r o n g s t a t e m e n t o f t h e o r e m 5 . 3 it i s a l r e a d y e a s y t o deduce a r e s u l t c o n c e r n i n g autD: 5.6.

THEOREM.

Given any b a l l B c c D c e n t e r e d a t a s D ,

t h e r e e x i s t s a c o n s t a n t K B s u c h t h a t we h a v e

70

CHAPTER

5

f o r a l l aeautD. I f B ' c c D is a n o t h e r b a l l c e n t e r e d a t a ' s D ,

ue have

on autD. P r o o f : Theorem 5 . 3 f u r n i s h e s a c o n s t a n t K

such t h a t

B

t

f o r a l l fsAutD. Hence, g i v e n AsautD and w r i t i n g f = : e x p t A r tdF, we have

tm.

But Tlim O't t o A a l s o i n t h e norms for a l l

I

t

( f - i d ) = A whence

11

- /IB

D

and

1

1 t

( 1 *:'I\

C

(ft-idD) tends

. This

proves (5.6).

s=o

To prove t h a t

w e need o n l y t o copy t h e proof of theorem 1 . 6 .

5.7.

* II

REMARK. The e q u i v a l e n c e of two norms on a v e c t o r

s p a c e i m p l i e s t h e e x i s t e n c e of p l

u1 I1 94.-

XI1

xi1

#

2 a l lX I 1

,

p 2 > 0 such t h a t w e have

1 for a l l x-

autD a s a Banach s p a c e .

A f t e r t h e p r e v i o u s theorem, t h e n e x t q u e s t i o n i s a t hand: Is autD endowed w i t h any o f t h e norms

I[ * \ I B

1

C 11 -:'I\ s=o c a u t D with or

a Banach

s p a c e ? . That i s , g i v e n a sequence ( A , ) 3 jm A!' +L(', s = 0 ,1 , does t h e r e e x i s t AeautD s u c h t h a t w e have la (

A,'=

L('

f o r s= 0,1? W e c a n prove a much s t r o n g e r r e s u l t t h a t

h a s c r u c i a l importance i n t h e t h e o r y of symmetric domains.

5.8.

THEOREM.

L e t t h e n e t s ( f j ) j e J C A u t D and

(tj), CIR: b e g i v e n a n d assume t h a t we c a n f i n d some acD, some JCJ such t h a t the n e t L(%E a n d some L ( ' ~ L ( E I E )

TOPOLOGY ON VECTOR FIELDS

A =:

1 t. 1

j

(f.-idD) I

s a t i s f i e s A ( s +L ( s f o r S = O , ? . jra t h a t we h a v e T l i m A , = A . jfJ

71

Then, t h e r e ezists AsautD

such

3

P r o o f : L e t u s f i x a b a l l B c c D c e n t e r e d a t acD and c h o o s e d>O s u c h t h a t B Z d c c D . We may assume

sup j E J

s=o

11

AiZll

<m

Then, by t h e o r e m 5 . 3 t h e r e e x i s t s M > O s u c h t h a t

11

fj-idDI/

6Mt 1

B2d

f o r a l l j c J . For 6 j o ( 6 ) , where M4 i s i n d e p e n d e n t of j , k, 8 a n d t . Applying t h i s r e s u l t t o ( 5 . 7 ) , t h e t r i a n g l e i n e q u a l i t y y i e l d s f n, l. - i d D r /

f kn - i d D k

1

n J. t J. A3.

t h a t is,

nktkAk

.

TOPOLOGY ON VECTOR FIELDS

73

f o r some M4>0, a l l j, k?j0(6) a n d a l l t e ( O , G / M ) .

Prom ( 5 . 9 ) it

r e a d i l y follows(by taking t h e superior l i m i t i n j , k with fixed that

6,t)

l i m sup11 j, k

A 3, - A ~ I ~ -~

< ~ ~ 6

i s a Cauchy n e t w i t h r e g a r d t o t h e ( A j )j e J C o n s e q u e n t l y , w e h a v e ( A , ) ( s + ~ ( s ' x ) , s= O , I , f o r

for a l l 6>0, i.e., norm

.

.

11

3 x

some ~ ( 0 t X ) e Eand L ( l ' X ) c L ( E - l E ) whenever xeB. But t h e n

i s a Cauchy n e t w i t h r e g a r d t o t h e norm

I( . \ I B ,

(A,)

I jeJ

whenever

i s a b a l l c e n t e r e d a t a p o i n t x s B . By r e p e a t i n g t h e

B'CCD

a r g u m e n t , f r o m t h e c o n n e c t e d n e s s of D w e o b t a i n t h a t ( A , ) , 3 IeJ i s a Cauchy n e t w i t h r e g a r d t o t h e norm / I . / I f o r any b a l l B"

B"CC

D,

(Aj)jeJ

i.e.

i s a T-Cauchy n e t . Thus w e h a v e

T l i m A = A f o r some A e H o l ( D , E ) . B u t t h e n , t h e o r e m 4 . 3 e s t a u i s h 3

e$ J t h a t AeatuD. 5.9.

COROLLARY.

L e t B c c D a n y baZZ c e n t e r e d a t aeD.

/ I .I1 B

T h e n a u t D is B a n a c h s p a c e w i t h r e g a r d t o t h e norms s=o

/ I "I;.

and

*

Proof: L e t us suppose t h a t ( A , ) , is a Cauchy s e q u e n I l a ce i n t h e norm /I .;*\I Then A!' * L ( s f o r s= 0 , l .

e

.

s=o

Choose a s e q u e n c e

(E,)

,

I l a

1la

o f p o s i t i v e numbers w i t h

E,*

I

0.

S i n c e w e have

1 T l i m - (exptA.-id ) = A . t I D I t+O

we can p i c k t . > O such t h a t 1

t . < E .

1

1

and

where f . = : e x p t A Obviously we have I j j'

J

f o r s= 0 , l . Now t h e o r e m 5 . 8 e n s u r e s t h a t L('=

A('

f o r some

74

CHAPTER

5

AsautD.

# 5.10.

EXERCISE. Prove c o r o l l a r y 5 . 9 d i r e c t l y by u s i n g

t h e l o c a l uniform c o n t i n u i t y o f t h e s o l u t i o n s o f o r d i n a r y d i f f g r e n t i a l equations with regard t o t h e i n i t i a l values.

5.-

autD a s a Banach-Lie a_ l g e_ bra. ~

L e t u s f i x any b a l l B c c D and 6>0 s u c h t h a t B 6 c c D , and endow

/I .I / B .

We a l r e a d y know t h a t ( a u t D , /I i s a Banach s p a c e . L e t u s now c o n s i d e r i t s L i e - a l g e b r a s t r u c t u re. autD w i t h t h e norm

5 . 1 1 . LEMMA. For a l l AcautD, t h e m a p p i n g A # : X+[A,X]

is a b o u n d e d Z i n e a r o p e r a t o r o n a u t D . P r o o f : The l i n e a r i t y of A f t i s o b v i o u s . On t h e o t h e r hand, by t h e Cauchy e s t mates and t h e f a c t

11 . l \ B ~ \ l

we

have

11

[A,X]

I/

=

/I A'lX-X'lA 1

~ ~ I I A l I lI X

BS

f o r a l l XeautD and some M ( i n d e p e n d e n t of X ) .

5 . 1 2 . COROLLARY. T h e m a p p i n g

# ; A+A#

is a c o n t i n u o u s

Z i n e a r o p e r u t o r on autD. P r o o f : W e have M I

11

A # ( l 6 M ' I I All

f o r a l l AcautD

and some

>O. 5.13.

PROPOSITION. We h a v e exp ( A # ) = (expA)

f o r a l l AcautD.

P r o o f : L e t XcautD be a r b i t r a r i l y f i x e d . By lemma 5.11

w e have t h a t

TOPOLOGY ON VECTOR FIELDS

[exp(tA ) ] X = : #

75

c t k AkX k! #

k=O

'L

i s a w e l l - d e f i n e d element Y ( t ) o f autD. Moreover,

d dt

?(t)= lim 1 "[ Y . ( t + h ) -".Y ( t ) ] = A Y % ( t ) # h+O h %

f o r a l l t6B a n d Y ( O ) = X . But t h e norm c o n v e r g e n c e of 1 [? ( t u h )-Y'L ( t )] means i t s T-convergence i n v i e w of c o r o l l a r y 5.12.

'L

Thus t h e mapping t + Y ( t )

s a t i s f i e s t h e d i f f e r e n t i a l equa-

tion (5.10)

i n t h e Banach space ( a u t D , T). But w e h a v e s e e n i n 5 4 C h a p t e r

IV, t h a t t h e mapping Y ( t ) = ( e x p t A ) # X ,

tm,

satisfies this

e q u a t i o n , t o o , whence t h e r e s u l t follows.

5.14.

LEMMA. L e t

#

@ be a c o n t i n u o u s automorphisrn of t h e

Banach L i e a 2 g e b r a a u t D . Then we h a v e

for a 2 2 A , XcautD. P r o o f : S i n c e @ i s a n automorphism of t h e L i e a l g e b r a autD, w e have

a n d , by r e i t e r a t i n g t h e a r g u m e n t w e o b t a i n

f o r n a . A s 4 i s a c o n t i n u o u s l i n e a r o p e r a t o r on a u t D , by p r o p o s i t i o n 5.13,

#

This Page Intentionaiiy Left Blank

CHAPTER

6

THE BANACH L I E GROUP STRUCTURE O F THE SET O F AUTOMORPHISMS

We have s e e n t h a t A u t D i s a t o p o l o g i c a l group when endowed w i t h t h e t o p o l o g y T of l o c a l uniform convergence. Now w e a r e g o i n 9 t o c o n s t r u c t anokher t o p o l o g y T a on AutD such t h a t (AutD, T a ) c a r r i e s t h e s t r u c t u r e of a r e a l Banach-Lie group which a c t s a n a l y t i c a l l y on D . F i r s t we i n t r o d u c e some p r e p a r a t o r y m a t e r i a l . The concept ___

51.-

of a Banach ___-__ manifold.

L e t M and E be r e s p e c t i v e l y a Hausdorff s p a c e and a Banach space o v e r any of t h e f i e l d s I R o r

which w e i n d i s t i n c t l y r e p r e s e n t

by X .

6 . 1 . D E F I N I T I O N . A " c h a r t V o f M o v e r E is a p a i r

(u,u)

w h e r e U i s a n o p e n s u b s e t of M a n d u is a h o m e o m o r p h i s m of U o n t o a n o p e n s u b s e t o f E.

on M i s a c o l l e c t i o n o f c h a r t s (Ual~ol)aeI E s u c h t h a t t h e foZZozJing c o n d i t i o n s a r e s a t i s f i e d :

An " a n a l y t i c s t r u c t u r e ' '

of M o v e r MI:

The f a m u l y

(Ua)aeI

i s a n o p e n c o v e r o f M.

M ~ : For e a c h p a i r a , ~ e ~ t h ,e m a p i n g l i B v v i l :

I - 1 , ( ~ ~ n ~ , ) +Bl(LJ i a

nu,)

is a n a l y t i c . M3:

T h e c o l l e c t i o n ( U a , ~ a ) a e I is a maximal f a m i l y o f c h a r t s on M f o r w h i c h c o n d i t i o n s M a n d M2hoZd. 1

A

" B a n a c h m a n i f o Z d " i s a p a i r ( M I A ) w h e r e M is a H a u s d o r f f

s p a c e a n d A is a n a n a l y t i c s t r u c t u r e o n M o v e r some B a n a c h s p a c e E. I f t h e r e i s no danger of c o n f u s i o n , w e s h a l l r e f e r t o t h e Banach manifold M w i t h o u t any r e f e r e n c e t o i t s a n a l y t i c s t r u c t u r e A .

77

CHAPTER

78

6

A c c o r d i n g a s t h e f i e l d x i s 3R o r .'U w e s a y t h a t M i s a p e a 2 o r a complex manifold. REMARK. C o n d i t i o n M 3 w i l l o f t e n be cumbersome t o c h e c k

6.2.

i n s p e c i f i c i n s t a n c e s . I n f a c t , i f c o n d i t i o n s M 1 and M2 a r e s a t i s f i e d , t h e f a m i l y (Uct,ucr)aer can be e x t e n d e d i n a unique manner t o a l a r g e r f a m i l y o f c h a r t s f o r w h i c h c o n d i t i o n M3 i s s a t i s f i e d , t o o . Thus, M3 i s n o t e s s e n t i a l i n t h e d e f i n i t i o n o f a Banach m a n i f o l d . 6 . 3 . EXEMPLES. L e t U b e a non v o i d o p e n s u b s e t of a Banach s p a c e E . The p a i r ( U , i d u ) i s a c h a r t o f U o v e r E a n d d e f i n e s a n a l y t i c s t r u c t u r e on U .

The m a n i f o l d so c o n s t r u c t e d i s c a l l e d

t h e c a n o n i c a i : m a n i f o l d on U . L e t M and N b e ,Banach m a n i f o l d s

F respectively. If

o v e r t h e Banach s p a c e s E a n d

( U , u ) a n d ( V , v ) are c h a r t s o f M a n d N , t h e n

(UxV, uxv') , where uxv: ( x , y ) + ( u ( x ) , v ( y ) ) , i s a c h a r t of M x N o v e r ExF. The f a m i l y o f t h e p a i r s So c o n s t r u c t e d i s a n

the pair

a n a l y t i c s t r u c t u r e and t h e c o r r e s p o n d i n g m a n i f o l d i s c a l l e d t h e

p r o d u c t of M and N . L e t M be a Banach m a n i f o l d o v e r a . c o m p l e x Banach s p a c e E. Then E c a n be c o n s i d e r e d a s a r e a l Banach s p a c e ,

by

%.

too, which we denote

Any c h a r t ( U , u ) o f M o v e r E i s a c h a r t o v e r

%

and t h e

f a m i l y o f t h e s e c h a r t s d e f i n e s a r e a l a n a l y t i c s t r u c t u r e on M . The m a n i f o l d so c o n s t r u c t e d i s c a l l e d t h e u n d e r Z y i n g r e a l

m a n i f o l d of M. 6.4.

DEFINITION. L e t a B a n a c h m a n i f o Z d M a n d a p o i n t xcM b e

g i v e n , a n d c o n s i d e r t h e s e t of t h e p a i r s

[ ( U , U ) rh] w h e r e

is a c h a r t of M at x and hsE. Me s a y t h a t [ ( V , U ) , h , ]

(U,u)

and

[ ( V , V ) , h 2 ] a r e " e q u i v a l e n t " i f We h a v e

( v O u - l )( '

u(x)

.h = h 1

2

W e w r i t e T M f o r t h e q u o t i e n t s e t . The e q u i v a l e n c e c l a s s o f t h e

a IX

e l e m e n t [(U,u) rh] which i s d e n o t e d by h au t a n g e n t v e c t o r t o M a t x.

,

is called a

THE L I E GROUP OF AUTOMORPHISMS

79

L e t u s f i x any c h a r t (U,u) of M a t xcM. The mapping E+T M g i v e n

a au. I

by h+ h

X

i s a b i j e c t i o n by means of which w e can

t r a n s f e r theXBanach s p a c e s t r u c t u r e of E t o T M. W e say that TJ4 X

endowed w i t h t h i s Banach s p a c e s t r u c t u r e i s t h e t a n g e n t s p a c e t o M a t x. 6.5.

DEFINITION. L e t a Banach m a n i f o Z d M and a Banach

space F be g i v e n .

We s a y t h a t a mapping f : M+F i s " a n a Z y t i c a t

a p o i n t xcM"if t h e r e i s a c h a r t f0u-I:

(U,u)

o f M a t x such t h a t

u ( U ) + F i s a n a z y t i c . We s a y t h a t f i s " a n a l y t i c o n M " if

i t i s a n a Z y t i c at e v e r y p o i n t xcM and we c a l l

f 0 u - l a "ZocaZ

expression" o f f a t x. L e t f, g: M+F be a n a l y t i c mappings a t a p o i n t XCM,

by f0u-l: u ( U ) + F , yov

-1

and d e n o t e

: v(V)+F t h e i r l o c a l r e p r e s e n t a t i o n s i n

t h e c h a r t s (U,u) and ( V , v ) , r e s p e c t i v e l y . W e s a y t h a t f and 9 a r e e q u i v a Z e n t a t x i f t h e r e i s a neighbourhood W c U f l V of x -1 -1 F on W . W e d e n o t e by B X t h e q u o t i e n t s e t such t h a t f o u = gav and each e q u i v a l e n c e c l a s s i s c a l l e d an a n a l y t i c germ a t x. O F

i s endowed w i t h a v e c t o r s p a c e s t r u c t u r e i n an o b v i o u s manner. NOW, t a n g e n t v e c t o r s t o M a t x can be i n t e r p r e t e d a s d i f k r e n t i a l

o p e r a t o r s a c t i n g on a n a l y t i c germs a t x i n t h e following manner: for f s O F

and h

a au

I x

eT M w e s e t x

a+ au 6.6.

( x ) .h=:

DEFINITION.

h (feu-')'' u (x)

L e t M and N be Banach m a n i f o l d s o v e r

t h e Banach s p a c e s E and F, r e s p e c t i v e l y . We s a y t h a t a continuous mapping f : M-+N i s a " m o r p h i s m " o f Banach m a n i f o l d s i f , for e a c h p o i n t XCM,

t h e r e are charts

y= f ( x ) s u c h t h a t v 0 f o u - l :

(U,u) of M a t x and

(V,v) of N a t

u ( U ) + v ( V ) is a n a l y t i c .

Suppose t h a t f : M+N i s a morphism of Banach m a n i f o l d s . Then (V.

€0

u -1)

(1

u (x)

i s an element of L ( E , F ) and w e c a n d e f i n e a

c o n t i n u o u s l i n e a r mapping d f ( x ) : T x M + T f ( x ) N by s e t t i n g

80

CHAPTER

6

for h e E . I t i s e a s y t o check t h a t df(x) does n o t depend on t h e c h a r t s ( U , u ) and (V,v) w e have chosen. W e say t h a t d f ( x ) i s t h e

d e r i v a t i v e of f a t x and t h a t ( 6 . 1 )

is i t s locuZ e x p r e s s i o n

w i t h r e s p e c t t o t h e c h a r t s ( U , u ) and ( V , v ) . L e t M be a Banach manifold o v e r E and l e t U b e a n open s u b s e t

of M. W e s e t iTxM; xcu)

TU=:

( U , u ) i s a c h a r t of M , w e d e f i n e a mapping T : TU+ u ( U ) x E

If

U

by means of

a

h -

T : u

au

Ix

f

(u(x), h ) .

Then, w e have 6.7.

PROPOSITION. T h e r e e x i s t s

cz

u n i q u e topology on TM

s u c h t h a t Lhc f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : ( a ) F o r eZ)Cray o p e n s u b s e t U of M, TU

Cn u n o p e n

:;uhsst o f TX.

(b) For1 e o c r y c h a r t ( U , u ) of M, T u : TU*u(U) X E is a h o m e o m o r p h i s m .

t h e mapping

W e l e a v e t h e proof a s an e x e r c i s e . I t i s c l e a r t h a t TM w i t h

t h i s topology i s a Hausdorff s p a c e . Moreover, if ( U , u ) i s a i s a c h a r t of TM o v e r t h e Banach space

c h a r t of M , t h e n (TU, Tu)

ExE, and w e have 6.8.

PROPOSITION. T h e f a m i l y { (TU,TU):

( u , u ) i s a c h a r t of M I

d e f i n e s a n d analytic s t r u c t u r e o n TM. The Banach m a n i f o l d so c o n s t r u c t e d on TM i s c a l l e d t h e tangent

-

b u n d l e t o M . Obviously, t h e c a n o n i c a l p r o j e c t i m s n 1' TM+M and TI TM-tE, g i v e n by

-

2'

a

nl: h au

Ix

+X

and

a v2: h -

au I x

+h

THE L I E GROUP O F AUTOMORPHISMS

a

81

cTM, are Banach m a n i f o l d morphisms. :x Moreover, i f f : M+N i s a morphism of Banach m a n i f o l d s , i t s

for h

I

d e r i v a t i v e d f : TM+TN i s a morphism o f t h e c o r r e s p o n d i n g t a n g e n t bundles.

6 . 9 . D C F I N I T ~ - O N . ~ ~ A ~ a n a l y t i c v e c t o r f i e l d ” o n a Banach m a n i f o l d M is morphism X: M+TM s u c h t h a t we have

I f X:

M+TM i s a n a n a l y t i c v e c t o r f i e l d on M , t h e n i t s v a l u e

a

Ix

X ( x ) a t xcM i s a t a n g e n t v e c t o r t o M a t x , X ( x ) = h ( x ) - c T M . au The l o c a l e x p r e s s i o n o f X w i t h r e s p e c t t o t h e c h a r t s (U,u) o f M a n d (TU,Tu) o f TM i s g i v e n by

where h: M-tE i s a n a n a l y t i c mapping on M . W e d e n o t e by T ( M ) t h e s e t o f a l l a n a l y t i c v e c t o r f i e l d s o n M .

6.10.

DEFINITION.Let X = :

f(x)

a ax

be a n a l y t i c v e c t o r f i e l d s o n M and l e t

l x and

Y= g ( x )

be g i v e n .

a

AX=: Af ( x ) au

a aulx

W e define

Ix

for xcM. It i s easy t o v e r i f y t h a t X+Y,

AX a n d [ X , Y ]

are elements of

T(M) a n d t h a t , i n t h i s way, T ( M ) becomes a L i e a l g e b r a . W e c a l l

it t h e L i e a l g e b r a of a n a l y t i c v e c t o r f i e l d s o n M . 6.11. manifolds.

DEFINITION. L e t @ :

Y c T ( N ) a r e “ r e l a t e d by

(6.2)

M+N be a morphism of Banach

W e say t h a t t h e a n a l y t i c v e c t o r f i e l d s XcT(M) and if we have

d$.X= Yo@

L e t u s t a k e c h a r t s (U,u) of M a t x a n d ( V , v )

o f N a t y = @ ( x ),

82

6

CHAPTER

and assume t h a t X = f ( x )

a au I x

and

a

Y= g ( y ) -

av IY

are t h e

c o r r e s p o n d i n g local e x p r e s s i o n s of X and Y . Then t h e e x p r e s s i o n

of

( 6 . 2 ) i s g i v e n by

6.12.

PROPOSITION.

L e t ip:

M+N be a m o r p h i s m o f Banach

m a n i f o l d s and assume t h a t X l I X 2 c T ( M ) a r e r e l a t e d b y (t, w i t h Y1,Y2cT(N),

r e s p e c t i v e l y . Then X1+X2,

r e l a t e d by $ w i t h Y 1 + Y z l h Y 1 and

AX1 and

are

[X1,X2]

[Y,,Y,].

W e leave t h e proof as an e x e r c i s e . 6.13.

DEFINITION.

L e t $: M+N b e a m o r p h i s m o f Banach

m a n i f o l d s . Then:

(a) We s a y thal ip i s an nirnmersion” i f , f o r e v e r y x c M , dip ( x ) : TxM*T

ip ( x )

N

i s i n j e c l i v e and t h e i m a g e d $ ( x ) . T M

c Z o s e d topologCcally c o m p l e m e n t e d s u b s p a c e o f T

is a

N.

(t, ( x )

( h ) We s a y t h a t @ i s a ” s u b m e r s i o n ” i f , for e v e r y x e M ,

dip(x) : T x M + T + ( x l N i s s u r j e c t i v e and t h e k e r n e l K e r d g ( x ) i s a ( o b v i o u s l y c l o s e d ) t o p o l o g i c a l l y complemented s u b s p a c e of T M. X

N o w we have ( s e e 12 I 5 5 ) . 6.14.

PROPOSITION. Let

4 : M+N be a m o r p h i s m of Banach

manifolds. Then the f o l l o u i n g statements are e q u i v a l e n t : (a) The mapping @ :

M-tN is a n i m m e r s i o n and a s u b m e r s i o n .

(b) For e a c h x c M , t h e mapping d i p ( x ) : TxM*T i p ( x j N is a

s u r j e c t i v e i s o m o r p h i s m of Banach s p a c e s .

( c ) For e a c h x c M , t h e r e a r e a n e i g h b o u r h o o d U of x in M and a n e i g h b o u r h o o d V o f y= $ ( x )

in N such t h a t

a n a l y t i c homeomorphism of U o n t o V. 6.15. ip:

D E F I N I T I O N . If a m o r p h i s m o f

@ IU

is a n

Banach m a n i f o l d s

M-tN s a t i s f i e s a n y o f t h e a b o v e c o n d i t i o n s , we s a y t h a t ip i s

a “ l o c a l i s o m o r p h i s m ” o f M and N.

83

THE L I E GROUP O F AUTOMORPHISMS

B y a n f ' i s o m o r p h i s m N o f Banach m a n i f o l d s we mean a b i j e c t i v e l o c a l isomorphism 6.16.

4:

M-tN.

PROPOSITION. L e t M , N and

4 be r e s p e c t i v e l y a

t o p o l o g i c a l s p a c e , a Banach m a n i f o l d o v e r E and a mapping

4 : M+N.

Then t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t ( a ) F o r e v e r y XCM,

t h e r e i s an o p e n n e i g h b o u r h o o d U o f

x i n M, t h e r e i s a c h a r t (V,v) o f y = : $ ( x ) in N and t h e r e is a c l o s e d t o p o l o g i c a l l y c o m p l e m e n t e d s u b s p a c e F of E s u c h t h a t v Q $ is a homeomorphism o f U o n t o F n vp# ( U ) ] . (b) T h e r e e x i s t s a Banach m a n i f o l d s t r u c t u r e o n M s u c h t h a t i t s u n d e r l y i n g t o p o l o g y i s t h e t o p o l o g y of M and

M+N

@:

i s an i m m e r s i o n . The m a n i f o l d s t r u c t u r e s a t i s f y i n g t h e s e c o n d i t i o n s i s u n i q u e and i t s c h a r t s a r e t h e p a i r s (V,V,I$

1") , where

U is as i n

(a)

.

W e c a l l it t h e @ - i n v e r s e image of t h e m a n i f o l d s t r u c t u r e i n N . 6.17.

DEFINITION. L e t N be a Banach m a n i f o l d and d e n o t e

by i : M-tN a t o p o l o g i c a l s u b s p a c e M o f N and t h e c a n o n i c a Z inclusion. I f the pair (M,i) s i t i o n 6 . 1 6 , we s a y t h a t M

s a t i s f i e s t h e c o n d i t i o n s of p r o p o endowed w i t h t h e i n v e r s e i m a g e

m a n i f o l d s t r u c t u r e o f t h a t in N is a s u b m a n i f o l d o f N . 52.-

The c o n c e p t o f a Banach-Lie

6.18.

group.

DEFINITION. A " B a n a c h - L i e " g r o u p is a s e t G w h e r e

we h a v e a g r o u p s t r u c t u r e t o g e t h e r Q i t h an a n a l y t i c s t r u c t u r e o v e r a Banach s p a c e E s u c h t h a t t h e mapping GxG+G g i v e n b y ( x , y ) + x y - l is a n a l y t i c . A c c o r d i n g a s E i s r e a l o r complex w e s a y t h a t G i s a r e a l o r a

compZex Banach-Lie

group.

I f e denotes t h e i d e n t i t y element o f G I we have

6.19.

PROPOSITION. L e t t h e s e t G

be endoved w i t h a

g r o u p s t r u c t u r e and an a n a l y t i c s t r u c t u r e o v e r E . T h e n G i s a Banach-Lie satisfied:

g r o u p i f and o n l y i f t h e f o l l o w i n g c o n d i t i o n s a r e

84

CHAPTER

6

L1: P'or nZli xOeG, t h e mapping G-tG g,Luen b y x+x x i s a n a l y t i c . 0

by

L 2 : For a1,l x0@G, t h c mapping G+G g i v e n

cinal'yt'ic .in a n o p e n neighbourhood

L3: T h e mapping GxG-+G o p e n n e i g h h o u r h o o d of

g i v e n by

x

+

x xx - 1

of e .

0

0

.is

(x,y)+xy-' i s a n a Z y t i c and

( e , e ).

P r o o f : I f G i s a Banach-Lie

group, t h e n t h e s e c o n d i t i o n s

a r e obviously s a t i s f i e d . Let

( x O t y 0 ) e G x Gb e g i v e n . Then w e h a v e

xy-i=

f o r a l l x,ysG.

~ x o y ; l ~ y o r ~ x ;( yl ox ~Y ) -1

-1

IY;'

Thus, t h e mapping ( x , y ) + x y - ' c a n be r e p r e s e n t e d

i n a n e i g h b o u r h o o d of

(x,,yo)

as a c o m p o s i t e of mappings o f t h e

t y p e s m e n t i o n e d i n c o n d i t i o n s L 1 , L2 a n d L

3'

whence t h e r e s u l t

follows.

7Y 6.20.

COROLLARY. L e t G b e e n d o w e d w i t h

a group s t r u c t u r e

and uri a n a 1 , y t i c s t r u c t u x a t . . T h e n G is a R a n u c h - L i e o n l y if t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :

group is a n d

L;:

T h e m a p p i n g G-tG g i v e n b y x+x-l i s a n a l y t i c o n G .

L;:

The mapping G x G + G g i v e n by

(x,y)+xy i s a n a l y t i c on GxG.

Proof: If ( x , y ) + x y - ' is a n a l y t i c , so a r e t h e m a p p i n g s y + ( e , y ) + e y - l and ( x , y ) - ( x , u - ' ) + x ( y - ' ) -

1

.

I f y-ty-l a n d

( x , y ) + x y a r e a n a l y t i c , so i s ( x , y ) + ( x , y - l ) + x y - l .

7Y 6.21. ~ P O U ~s tJ r

COROLLARY. L e t G b e a B a n a e h - L i e

group. T h e n the

u c t u r e o n G is c o m p a t i b Z e w i t h t h e t o p o Z o g y u n d e r Z y i n g

t h e a n a l y t i c s t r u c t u r e o$ G , 6.22.

i.e.,

G i s a topoZogicuZ group.

EXERCISES. Assume t h a t G i s a Banach-Lie

group.

85

THE LIE GROUP OF AUTOMORPHISMS

Show that the topological group G satisfies the following conditions: (a) G is metrizable. (b) Both the left and right uniform structures of G are complete. 6.23. DEFINITION. L e t G a n d H b e B a n a c h - L i e g r o u p s . We Q m a p p i n g f: G+H is a " m o r p h i s m " of B a n a c h L i e g r o u p s

say t h a t

if f is a m o r p h i s m of b o t h t h e g r o u p s t r u c t u r e s a n d t h e manifold s t r u c t u r e s of G a n d H. 6.24. PROPOSITION.Let G a n d H b e Banach-Lie groupsand denote b y f: G+H a g r o u p h o m o m o r p h i s m . T h e n f i s a morphism of

Banach-

L i e g r o u p s if a n d o n l y i f f: G-tH i s a n a l y t i c i n a n e i g h b o u r h o o d o f e.

Proof: Let x sG be given. If f: G+H is a group homomor0 phism, we have f(x)= f(xilx) for all xcG. Thus, by conditions L' and L; of corollary 6.20,if f is analytic in a neiqhbourhood 1 of e, it is analytic on G. The converse is obvious. 6.25. DEFINITION. L e t t h e B a n a c h - L i e g r o u p G a n d t h e e l e m e n t asG b e g i v e n . We d e f i n e t h e r r Z e f t " a n d " r i g h t t r a n s l a t i o n s " b y a a s t h e m a p p i n g s G+G g i v e n r e s p e c t i v e l y b y

La: x+ax

r

:

x+xa

,

xeG.

Obviously, La and ra are automorphisms of the analytic structure of G. Moreover, the mapping i

:

x+axa

-1

xsG

is a Banach-Lie group automorphism of G. 6.26. DEFINITION. L e t G b e a B a n a c h - L i e g r o u p . We s a y t h a t a s u b s e t H c G i s a " B a n a c h - L i e s u b g r o u p " of G is H is a s u b g r o u p a n d a s u b m a n i f o l d of G w i t h r e s p e c t t o t h e c a n o n i c a l i n c l u s i o n i: H-tG.

86

6

CHAPTER

6.27.

EXERCISES.

( a ) L e t G be a B a n a c h - L i e

t h a t t h e i d e n t i t y component of e i s a B a n a c h - L i e ( b ) L e t H be a B a n a c h - L i e G.

g r o u p . Show subgroup of G. s u b g r o u p of

Show t h a t H i s c l o s e d a n d t h a t t h e c a n o n i c a l i n c l u s i o n

i : H-tG i s a morphism o f B a n a c h - L i e

6.28.

DEFINITION.

groups.

L e t G be n B a n a c h - L i e

g r o u p . We sag

thal an a n a l y t i c u e c l v r f i e i ' d X c T ( G ) i s " l e f t i n v a r i a n t " ,if, for all acG, X i s r e l a l e d t o i t s e l f b y Ra , i.e., i f ZJC h a v e (6.3)

dQ,;X=

aeG

XoK.

a

is t h e local ( U , u ) i s a c h a r t of G a n d X = f ( x ) au / x e x p r e s s i o n of X I t h e n ( 6 . 3 ) i s e q u i v a l e n t t o If

W e d e n o t e by G ( G ) t h e s u b s e t of Y ( G ) c o n s i s t i n g of a l l l e f t

i n v a r i a n t a n a l y t i c v e c t o r f i e l d s on G. A s a n immediate co n s e-

we obtain

q u e n c e of p r o p o s i t i o n 6 . 1 2 , 6.29.

PROPOSITION. L e t G be u Banach-Lie g r o u p .

G ( G ) is a L i e s u b a l g e b r a of

6.30.

Then

T(G).

PROPOSITION. L e t

(Y:

G ( G ) + T e ( G ) b s t h e evaZuaLion

at I h e p o i n t eeG. T h e n a is a s u r j e c t i v e isomorphism of v e c t o r spuces. P r o o f : L e t (U,u) be a c h a r t o f G a t e ; t h u s

by X - + X ( e )

a au

le

f o r XcG(G)

b e c a u s e of d e f i n i t i o n 6 . 1 0 .

some X,YeG(G).

. Clearly

c1

is given i s a l i n e a r mapping c1

Assume t h a t w e h a v e X ( e ) = Y ( e ) f o r ( e )I w e

A s X a n d Y are l e f t i n v a r i a n t and a = R

have

for a l l acG.

Let h

a au

le

Thus X= Y and a

is i n j e c t i v e .

e T e ( G I b e g i v e n . Then w e d e f i n e X ( a ) =: dR (e) .h

a

au

le

87

THE L I E GROUP O F AUTOMORPHISMS

f o r asG a n d it i s i m m e d i a t e t o c h e c k t h a t dRe ( e ): Te ( G ) + T e ( G )

i s t h e i d e n t i t y mapping. Thus

Moreover, X i s a n a l y t i c . Indeed, s i n c e l e f t t r a n s l a t i o n s a r e a u t o m o r p h i s m s of t h e m a n i f o l d s t r u c t u r e o f G ,

(aU, u 0 R - l )

c h a r t o f G a t t h e p o i n t a and t h e l o c a l e x p r e s s i o n of R

a

is a is the

i d e n t i t y map. T h u s , X i s l o c a l l y r e p r e s e n t a b l e a s t h e c o n s t a n t mapping x-th

a au

(x

f o r xsaU a n d X i s a n a l y t i c . B e s i d e s , X i s

l e f t invariant since

X [ R a ( x ) ] = X ( a x ) = dR

ax

(e).h

a au

le

=

f o r a l l a,xsG. T h e r e f o r e , l e f t i n v a r i a n t v e c t o r f i e l d s on a Banach L i e g r o u p a r e a n a l y t i c a n d t h e y a r e u n i q u e l y d e t e r m i n e d by t h e i r v a l u e s a t the point esG. By means of t h e i s o m o r p h i s m a: G ( G ) + T e ( G )

we can t r a n s f e r t h e

Banach s p a c e s t r u c t u r e o f T e ( G ) t o G ( G ) , a n d i t i s i m m e d i a t e t o v e r i f y t h a t , i n t h i s way, G ( G ) becomes a Banach-Lie a l g e b r a . W e c a l l i t t h e Bannck-Lie

53.-

a Z g e b r a of G .

S p e c i f i c ____ e x a m p l e s : The l i n e a r g r o u p a n d i t s a l g e b r a i c subgroups.

L e t A b e a r e a l o r complex Banach a l g e b r a w i t h u n i t e . W e

i n d i s t i n c t l y d e n o t e b y M a n y o f t h e f i e l d s IR o r it.

6.31.

DEFINITION. We d e f i n e t h e " c o m m u t a t o r p r o d u c t " o n

A by m e a n s o f

[x, y] = : xy-yx

x,ycA.

CHAPTER

88

6

I t i s i m m e d i a t e t o see t h a t t h i s p r o d u c t s a t i s f i e s t h e c o n d i -

t i o n s o f d e f i n i t i o n 4 . 2 0 a n d t h a t t h e commutator p r o d u c t

[,I:

AxA-tA

i s c o n t i n u o u s . Thus A i s a Banach-Lie a l g e b r a .

L e t u s d e n o t e by G ( A ) t h e s e t o f r e g u l a r e l e m e n t s o f A ; G(A)

thus,

i s a g r o u p and a n o p e n s u b s e t of A . T h e r e f o r e , G ( A ) i s a

Banach m a n i f o l d i n a c a n o n i c a l manner ( c f . e x a m p l e s 6 . 3 ) .

Now

w e have

6 . 3 2 . LEMMA. W i t h ils c a n o n i c a l structures of g r o u p a n d g r o u p #hose B a n a c h - L i e

B a n a c h manifold, G ( A ) is a B a n a c h - L i e

algebra is A . p r o o f : I t i s i m m e d i a t e t o c h e c k t h a t c o n d i t i o n s L 1 and L

of p r o p o s i t i o n 6 . 1 9 a r e s a t i s f i e d . F o r yeA w i t h 2 we h a v e

)I

y-eI\ < I

m

Y-l=

[ e + ( y - e ) J -l;

x

1"

(-

n=O

t h e series b e i n g c o n v e r g e n t i n t h e norm of A . (x,y)+xy-l i s a n a l y t i c i n a neighbourhood of

T h u s , t h e mapping (e,e), i.e.,

c o n d i t i o n L 3 i s s a t i s f i e d , t o o , a n d G ( A ) i s a Banach L i e g r o u p o v e r t h e Banach s p a c e A. Let a c G ( A ) be f i x e d ; w i t h r e s p e c t t o t h e c a n o n i c a l c h a r t , t h e expression of t h e l e f t t r a n s l a t i o n R a i s R T h u s , i t s d e r i v a t i v e dR

a

( X I - ax f o r x c G ( A ) .

i s g i v e n by

I f X= X(x)

3 -

is a l e f t invariant au v e c t o r f i e l d on G ( A ) , by d e f i n i t i o n 6 . 2 8 w e have f o r a l l xeG(A) a n d hwl.

I

dR (x).X ( x ) = X [ R a ( X I By ( 6 . 4 )

]

t h i s is equivalent t o

f o r a l l a,xeG(A). Taking x= e we o b t a i n X ( a ) = a X ( e ) f o r a c G ( A ) o r , by c h a n g i n g t h e n o t a t i o n ,

THE LIE GROUP O F AUTOMORPHISMS

89

xcG(A)

(6.5)

I t i s e a s y t o see t h a t v e c t o r f i e l d s of t h e

where h = : X ( e ) c A .

form ( 6 . 5 ) a r e a c t u a l l y l e f t i n v a r i a n t . M o r e o v e r , f o r X=:

xh

a

1

z/ x

and

a

xh2

Y=:

[ X , Y ] ( x ) = ( x h h -xh h 2

1

lX

1 2

a au

w e have

Ix

= X[hl,h2]

a au lX

so t h a t t h e mapping T G ( A ) + A o b t a i n e d by e v a l u a t i n g a t e c G ( A ) i s a s u r j e c t i v e Banach-Lie

isomorphism between T G ( A ) and A .

# 6.33.

DEFINITION. We s a y t h a t t h e B a n a c h - L i e

group G ( A )

i s t h e "Zineari g r o u p " o f t h e Banach a l g e b r a A a n d d e n o t e i t b y GL(A;IK).

Assume t h a t A i s complex Banach a l g e b r a ; t h e n it i s a r e a l

.

Banach a l g e b r a , t o o , w h i c h i s d e n o t e by A Thus t h e l i n e = IR group G L ( A , E ) , with i t s underlying r e a l manifold s t r u c t u r e , i s a r e a l Banach-Lie

G.W e

group over

s a y t h a t it i s t h e underZying

reaZ l i n e a r g r o u p o f G L ( A , E ) .

6.34. DEFINITION. For t d R and x c A exptx=:

we d e f i n e

t n xn

C n! n=O

S i n c e A i s c o m p l e t e , w e h a v e e x p t x c A ; a c t u a l l y , e x p t x i s a reg u l a r e l e m e n t of A a n d ( e x p t x ) - I = e x p ( - t x )

,

so t h a t t h e mapping

e x p : I R x A + G L ( A J K ) g i v e n by ( t , x ) + e x p t x i s r e a l a n a l y t i c . F o r t = l

w e s i m p l y w r i t e e x p x i n s t e a d o f e x p l x . The mapping A + G L ( A , I K ) g i v e n by x+expx i s r e a l a n a l y t i c , too. 6.35. @:

PROPOSITION. L e t A , B b e Banach a Z g e b r a s and

G L ( A J K l + G L ( B , I K ) a m o r p h i s m of t h e c o r r e s p o n d i n g

Z i n e a r groups

Then t h e d e r i v a t i v e d @( e ) : TeGL ( A ,lK) +TeGL ( B & ) i s a homomorphism o f t h e B a n a c h - L i e a l g e b r a s A and B and we h a v e @ ( e x p x ) = e x p [d@( e )x]

CHAPTER

90

6

Proof: For any fixed xeA, the mapping f: I R + G L ( B ; I K ) given by t+f(t)=: b(exptx1 is real analytic. Moreover, since $ is a group homomorphism, by setting a=: d$(e)xsB, we have f ( 0 ) =$(e)=e and

=

CP (exptx)lim g1 [+(expsx)-el= $(exptx)a s+o

Thus,f is the solution of the initial value problem

in the Banach space B. Now, we consider the function g: I R + G L ( B , l K ) given by t+g(t)=: expta with a= d@(e)x. It is easy to see that g(0)= e and

Thus g is also the solution of (6.6) and we have f(t)= g(t), i.e. $(exptx)= exptd$(e)x for all tdIR and xcA. NOW, let x,ycA be given and consider the mapping F : I R + G L ( B J K ) given by F(t)=: $(exptx)$(expty), tcTR. A computation similar to the one above gives ( 6-7)

F' (t)= $ (exptx) (a+b)@ (expty)

where we have put a= d@(e)x, derivative of (6.7) at t= 0

b=: dQ(e1y. By taking the

F" ( 0 ) = aL+2ba+bL

Similarly, if G(t) = :

+ (expty)@ (exptx), we

have

THE L I E GROUP O F AUTOMORPHISMS

91

2 2 G " ( O ) = a +2ab+b

so t h a t Y ( t )= : F ( t ) - G ( t )

Also,

satisfies

a p p l y i n g twice t h e c h a i n r u l e a t t = 0 t o compute Y " ( 0 )

we o b t a i n

whence t h e r e s u l t f o l l o w s by comparing w i t h ( 6 . 8 ) .

6.36.

EXERCISE. L e t '4:

A+B b e a Banach-Lie

algebra

homomorphism. Show t h a t t h e r e e x i s t s a unique Banach-Lie homomorphism such t h a t d $ ( e ) = Y . 6.37.

PROPOSITION. L e t H b e a s u b g r o u p o f t h e l i n e a r

g r o u p GL(A,X) a n d l e t B b e a c l o s e d s u b s p a c e of A . A s s u m e t h a t t h e r e a r e a n e i g k b o u r k o o d U of e i n G L ( A , l K ) a n d a n e i g k b o u r k o o d V of

0 i n A s u c k t h a t t h e e x p o n e n t i a l m a p p i n g exp: V n B + U n H i s

a homeomorphism f o r t h e c o r r e s p o n d i n g i n d u c e d t o p o l o g i e s . T k e n B i s a B a n a c k - L i e s u b a l g e b r a of A a n d H i s a B a n a c k - L i e w h o s e L i e a l g e b r a i s B.

group

P r o o f : S i n c e exp ( '= i d , by t h e i n v e r s e mapping theorem 0

t h e r e i s no loss of g e n e r a l i t y i n assuming t h a t ( U , l o g

) IU

,

is a

c h a r t of G L ( A , I K ) a t e l where log d e n o t e s t h e i n v e r s e of exp. Thus ( U n H , 1 0 g l U n H i s a c h a r t of H o v e r t h e Banach s p a c e B . Now, l e t hcH be g i v e n . A s H i s a subgroup of G L ( A , I K ) t r a n s l a t i o n R h maps U l l H o n t o a s e t R , ( U n H ) c H

,

the l e f t

which i s a

neighbourhood of h i n H I and t h e p a i r (6.9) i s a c h a r t of H a t h . I t i s e a s y t o see t h a t t h e f a m i l y g i v e n by ( 6 . 9 )

f o r hcH i s an a n a l y t i c s t r u c t u r e o v e r B . Moreover, f o r

t h i s a n a l y t i c s t r u c t u r e , H i s a Banach-Lie p o s i t i o n 6.30,

B i s a L i e s u b a l g e b r a of A .

group; t h u s by pro-

#

CHAPTER

92

6.30. nach-Lie

6

REMARK. N o t i c e t h a t , i n g e n e r a l , H i s n o t a Ba-

s u b g r o u p o f GL(A,IK) b e c a u s e , a s a m a n i f o l d , H may f a i l

t o be a s b n i f o l d o f GL(AJK): i t s t a n g e n t space a t e is B , which i n g e n e r a l i s n o t a complemented s u b s p a c e o f A . 6.39.

DEFINITION. L e t A b e a Banach a l g e b r a o v e r I K and

l e t H he a s u b g r o u p of G L ( A J K ) . We s a y t h a t H is ::ithgroup of d e g r e e Cn of GL(A,IK) S

(I

IK-algebraic

if t h e r e e z , i s t s a n o n v o i d s e t

of c o n t i n u o u s v e c t o r - U a Z u e d I K - p o l y n o m i a Z s q : AxA-tE

cn w i l h q ( O , O ) =

of

degree

0 s u c h t h a t w e have

Of p a r t i c u l a r i n t e r e s t f o r u s , t h o u g h n o t i n c l u d e d i n t h e a b o v e d e f i n i t i o n , i s t h e s i t u a t i o n i n w h i c h w e h a v e a Banach a l g e b r a o v e r C and a s u b g r o u p H o f G L ( A , t ) , ' b u t t h e e q u a t i o n s ( 6 . 1 0 ) d e f i n i n g H a r e ' l l i - p o l y n o m i a l s q: AxA+E o n t h e u n d e r l y i n g El-struct u r e s of A x A

and E

Notice t h a t , i n a l l t h e s e c a s e s , H is closed i n GL(AJK).

Clear-

l y , a n y f i n i t e p r o d u c t a n d a n y i n t e r s e c t i o n of IK-algebraic s u b g r o u p s o f d e g r e e & n i s a M - a l g e b r a i c s u b g r o u p of d e g r e e Cn. The d e f i n i t i o n i n c l u d e s t h e c a s e i n which H i s d e f i n e d b y a s e t S o f M - p o l y n o m i a l s q : A+E d e p e n d i n g on a s i n g l e v a r i a b l e . A l s o ,

by t h e Hanh-Banach t h e o r e m , t h e p o l y n o m i a l s qcS can b e c h o s e n t o b e IK-valued. 6.40.

ussume t h a t H

THEOREM. L e t A b e a Bunach a l g e b r a o v e r I K and

i s aIK-algebraic

s u b g r o u p of d e g r e e hn of

G L ( A , I K ) . Then H is u Banach L i e g r o u p whose B a n a c h - L i e

algebra

Is

P r o o f : W e w r i t e w=: ( u , v ) f o r t h e e l e m e n t s o f AxAvihich i s a Banach a l g e b r a o v e r I K w i t h r e s p e c t t o t h e norm

/I w J / =:

M-Banach

, //

n

v / / I . L e t u s p u t P = : 9 ~ ~ ( a x . 4f )o r t h e k=l s p a c e o f continuouslK-polynomials p: AxAjlK of d e g r e e

max{// u l /

s n s u c h t h a t p ( O , O ) = 0 . Now w e d e f i n e a mapping

THE L I E GROUP O F AUTOMORPHISMS

@:

93

GL(AJK)+GL(P(P) ,K) by means of [ @ ( x ) p(]u , v ) = : p ( u x , x - l v )

where pcP, Y : A+L(P)

(u,v)eAxA and xcGL A S ) . A l s o , w e d e f i n e a mapping by [Y(x)p] ( u , v ) =:

where pcP,

(u,v)cAxA and xcA.

F i r s t , w e s t u d y some p r o p e r t i e s of 0 and Y . W e have

( a ) 4 i s a Banach-Lie group homomorphism and d + ( e ) = Y. The proof i s a n e x e r c i s e . Thus, by p r o p o s i t i o n 6 . 3 5 w e g e t ( b ) Y i s a Banach-Lie

a l g e b r a homomorphism and

[ @ ( e x p x ) p ] =[ ~ X P + ( X ) ] P

(6.11)

f o r a l l xcA and pep. ( c ) L e t z s A be g i v e n . Then e a c h of t h e s u b s p a c e s P k ( ~ x A ) k, = l , 2 , . . , n r

i s i n v a r i a n t by Y ( z ) , i . e . ,

w e have Y ( z ) c P k ( A x A ) . Moreover, i f z s A i s a r e g u l a r e l e m e n t of A , t h e n Y ( z ) i s a r e g u l a r e l e m e n t of L ( P ) . I n d e e d , l e t pcP(AxA) be g i v e n and suppose t h a t F c L k ( A x A , I K ) i s

its a s s o c i a t e d symmetric k - l i n e a r mapping s o t h a t w e have

Then, t h e mapping f : AxA-+AxA g i v e n by (6.12)

f ( z ) : w= ( u , v )

+

(uz,-zv)

s a t i s f i e s f ( z ) c L ( A x A ) . T h e r e f o r e , from t h e d e f i n i t i o n of y w e get

and Y ( z ) pcPk ( A K A ) .

Now, suppose t h a t zcA i s r e g u l a r . Then f

( 2 ) a s d e f i n e d by ( 6 . l a i s r e g u l a r i n L ( A x A ) and f ( z ) - l = f ( z - I ) . Thus, t h e r e s t r i c t i o n

94

6

CHAPTER

o f Y ( z ) t o each of t h e s u b s p a c e s P ( A x A ) i s a r e g u l a r element k o f L(Pk(AxA)) , t h e i n v e r s e image of pePk(AXA) b e i n g g i v e n by

Therefore Y ( z ) i s r e g u l a r i n L ( P )

,

too.

Next w e show t h a t B i s a c l o s e d L i e s u b a l g e b r a of A . S i n c e H i s a IK-algebraic subgroup of d e g r e e s n of G L ( A , X ) t h e r e i s a s e t of IK-polynomials

,

S c P such t h a t

H= I ~ ~ G L ( A ; I K ) ;q ( z , z - ' ) =

o

VqcS 1

W e d e f i n e a n o t h e r s e t o f polynomial Q c P by means of

(6.13)

Q=:

{pep;

p(h,h-')= 0

VhcH}

C l e a r l y , Q i s a c l o s e d M - s u b s p a c e of P and S c Q : t h u s , i n particular (6.14)

[zcGL(AJK), q ( z , z - ' ) = 0

VqcQ]=>

zeH

We c l a i m t h a t , f o r xeGL(AJK), w e have t h e e q u i v a l e n c e (6.15)

xeH < = > + ( x ) Q c Q

Indeed, l e t xeGL(A,X) be g i v e n and assume t h a t xsH. A s H i s a subgroup of G L ( A , l K )

,

we have Hx= H-lx= H . From ( 6 . 1 3 ) and t h e

d e f i n i t i o n of 4 w e o b t a i n

f o r a l l qcQ and hcH; t h u s @ ( x ) Q C Q by ( 6 . 1 3 ) .

Conversely, l e t

xeGL(A,X) be g i v e n and assume t h a t + ( x ) Q c a . By ( 6 . 1 4 )

it

1

s u f f i c e s t o show t h a t q ( x , x - ) = 0 f o r a l l qcQ. L e t qeQ be given; by assumption we have @ ( x ) q c Q t; h u s by ( 6 . 1 3 ) w e o b t a i n -1 - 1 i . e . , q ( h x , x h ) = 0 for a l l heH, and [ @ ( x ) q ] ( h r h - l ) =0 , t a k i n g h= e s H we g e t q ( x , x - l ) = 0 . Now w e c l a i m t h a t , f o r yeA, w e have t h e e q u i v a l e n c e

95

THE L I E GROUP OF AUTOMORPHISMS

y€B < = > Y ( y ) Q c Q

(6.16)

I n d e e d , l e t ysB be g i v e n . Then we have exptycH f o r a l l tdR a n d , by (6.15)

,

@ ( e x p t y ) Q c Q t; h e r e f o r e , from ( 6 . 1 1 ) w e d e r i v e

[ e x p Y ( y ) ] Q C Q .I f w e f i x any q c Q , t h e mapping IR-tP g i v e n by t-texpY(y)q t a k e s i t s v a l u e s i n t h e closedIK-subspace Q of P a n d , by t a k i n g i t s d e r i v a t i v e a t t = 0 , w e g e t Y!(y)qcQ, whence Y ( y ) Q c Q . Conversely,

l e t yeA be such t h a t Y ( y ) Q c Q . A s Q i s

a c l o s e d x - s u b s p a c e of P , w e have [ e x p t + ( y ) ] Q c Qf o r a l l

tm.

Thus, by ( 6 . 1 1 ) , @ ( e x p t y ) Q c Q ,whence exptysH f o r a l l tdR and t h e r e f o r e ycB. I n p a r t i c u l a r , a s d @ ( e ) =Y i s a Banach-Lie a l g e b r a homomorphism,

( 6 . 1 6 ) e n t a i l s t h a t B i s a c l o s e d L i e s u b a l g e b r a of A . Next, w e show t h a t H i s a Banach-Lie group.

a:

L e t A =: A Q i A & the complexified of (I:

A = A when

IK= a)

. For

xcA

(I:

,

t h e Banach a l g e b r a A ( t h u s

S p ( x ) 1s the spectrum of x

i n A'.

From

t h e s p e c t r a l t h e o r y we know t h a t t h e s e t s

and

a r e , r e s p e c t i v e l y , neighbourhoods of e i n G L ( A J K ) and 0 i n A . According t o t h e holomorphic f u n c t i o n a l c a l c u l u s ( c . f .

11

I ) , on

U w e can s e l e c t a holomorphic b r a n c h of t h e l o g a r i t m i c f u n c t i o n

L e t u s d e n o t e by l o g i t s p r i n c i p a l d e t e r m i n a t i o n . By t h e spectral

mapping theorem, l o g : U+V i s a complex b i a n a l y t i c map ( t h u s , a r e a l b i a n a l y t i c map, t o o , i n c a s e I K = I R ) whose i n v e r s e i s exp: V+U.

T h e r e f o r e , by p r o p o s i t i o n 6 . 3 7 ,

it s u f f i c e s t o show

t h a t we have exp ( V

nB ) c U

H

log(U n H ) c V f l B

NOW, l e t ycV f l B be g i v e n ; from ycV and ycB w e g e t expycu and

exptycH f o r a l l tm, t h u s expycU n H .

CHAPTER

96

6

Next, l e t xcUnH be g i v e n and p u t y=: l o g x . Thus, i n p a r t i c u l a r (6.17)

YCV

M o r e o v e r r f r o m XSU w e d e r i v e

a n d by t h e s p e c t r a l mapping t h e o r e m

W e c l a i m t h a t t h e s p e c t r u m Sp(TY(y)] of ' Y ( y ) i n t h e c o r n p l e x i f i e d algebra L(P)'

of L ( P ) ( o r i n L ( P ) whenlK= C) s a t i s f i e s

Indeed, w e p r o v e . t h a t f o r A r e g u l a r element of L ( P )

'.

d with

(imgX/2v

I

AI-Y(y) i s a

Now w e h a v e

so t h a t

f o r pePk(AxA), k = 1 , 2 , . . , n ,

a n d w= ( u , v ) e A x A . T h u s , it s u f f i c e s

A

i; I - f ( y ) i s r e g u l a r i n L ( A x A ) f o r k = 1 , 2 , . . , n . B u t , due t o t h e d e f i n i t i o n of f ( y ) ,

t o prove t h a t

S i n c e by ( 6 . 1 8 ) ,

I imgh I ? v / n

e n t a i l s h/kgSp ( y ) a n d A/kgSp ( - y ) A A e-y a n d i; e + y a r e r e g u l a r i n

f o r k= I , 2 r . . r n r t h e e l e m e n t s A a n d so i s

Since

x k

I-f ( y ) i n L (AxA)

4 ( x )= 4 ( e x p y ) =

expY ( y )

.

by t h e f u n c t i o n a l c a l c u l u s a n d

t h e s p e c t r a l mapping t h e o r e m i t f o l l o w s t h a t

97

THE L I E GROUP O F AUTOMORPHLSMS

By R u n g e ' s t h e o r e m , t h e r e i s a s e q u e n c e of p o l y n o m i a l s p k c C \ X / , kdN, s u c h t h a t w e have l o g h = limpk(X) u n i f o r m l y when XcSp I @ ( x ) 1

. Then

k+m

I

1 Y ( y ) = log$ ( x ) = [Xe-Q( x ) ]-IlogAdX= 2 ~ i JY

S i n c e xeH, by ( 6 . 1 5 ) w e have @ ( x ) Q C Q a n d , as Q i s a closed s u b s p a c e of PI w e g e t Y ( y ) Q = l i m p k [ + ( x ) ] Q C Q .Then, by ( 6 . 1 6 ) w e a

obtain

k+m

Y ( y )c B

(6.19)

F i n a l l y , from ( 6 . 1 7 ) and ( 6 . 1 9 ) w e deduce Y ( y ) c V n B .

ff 6.41. over

a:

REMARK.

T h e case i n which A i s a Banach a l g e b r a

and H i s a n n - a l g e b r a i c

included i n our considerations.

subgroup of GL(A,E)

c a n a l s o be

Indeed, we can c o n s i d e r t h e

u n d e r l y i n g = - s t r u c t u r e s of A and G L ( A , C ) and d e f i n e n P = @ P ( A x A ) t o b e t h e Banach s p a c e of c o n t i n u o u s l R - p o l y n o m i a l s k=1

P: AxA

k

-t

a:

of d e g r e e Sn w i t h p ( O , O ) = 0 . A s i n o u r c a s e t h e

polynomials defining H belong t o a subset of P I we a r e i n a s i t u a t i o n i n which t h e o r e m 6 . 4 0 i s a p p l i c a b l e . A number o f i n t e r e s t i n g examples o f X - a l g e b r a i c

s u b g r o u p s of

d e g r e e s n of GL(A,IK) are i n c l u d e d i n t h e f o l l o w i n g PROPOSITION. L e t X I Y and f s L ("X,Y)

6.42.

be r e s p e e t i v e z y

t w o Banach s p a c e s 0 v e r . X and a x - r n u Z t i Z i n e a r mapping f : Xx...xX-+Y

.

Let

m= 0 o r m= 1 and s u p p o s e t h a t X= Y when m= 1 .

Then, t h e s e t H o f t h e eZements clcGL(L(X),IK)

satisfying

98

CHAPTER

f

(6.20)

( a x 11 . .

,(YX

n

1 = am f

(XI

I

-

6

* lXn)

XI

1 . .

,x

cx

is u B a n a c h - L i e g r o u p ~ l h o s eBanach-Lie a l g e b r a B is t h e s e t of a1 1. 6 c L ( X ) s a t i s f y i n g f(6x

(6.21)

1

,..,x n ) +..+

f(xl,...,fixn ) = m d f ( x l ,.., x n )

Here am a n d mfi d e n o t e r e s p e c t i v e l y t h e i d e n t i t y a n d t h e z e r o t r a n s f o r m a t i o n on Y when m= 0 . P r o o f : O b v i o u s l y A = : L ( X ) i s a Banach a l g e b r a o v e r K a n d H I a s d e f i n e d by ( 6 . 2 0 ) GL ( L(X)JK)

. We

i s a subgroup of t h e l i n e a r group

c l a i m t h a t H i s a IK-algebraic

s u b g r o u p of d e g r e e

Cn. I n d e e d , f o r f i x e d x 1 , x 2 , . . x ex, t h e mapping : T,(X)+X

PXlI.. lxn

g i v e n by

i s a c o n t i n u o u s n-homogeneous IK-polynomial,

and H i s d e f i n e d by

t h e s e t S of e q u a t i o n s PXlI..,X

( a ) =0

f o r x l , . . , x n ~ X . M o r e o v e r , B a s d e f i n e d by ( 6 . 2 1 ) ,

Banach-Lie

is a closed

s u b a l g e b r a of A = L ( X ) . T h u s , it s u f f i c e s t o show

t h a t H and B are a s i n theorem 6 . 4 0 i . e . w e have B= ( n e L ( X ) ; e x p t a c H

Vtm]

NOW, s u p p o s e t h a t G e L ( X ) s a t i s f i e s e x p t 6 e H f o r a l l

R e p l a c i n g cx b y e x p t d i n ( 6 . 2 0 )

tm.

and t a k i n g t h e d e r i v a t i v e a t

t = 0 w e see t h a t 6 s a t i s f i e s ( 6 . 2 1 ) ; t h u s 6 c B . C o n v e r s e l y , s u p p o s e t h a t 6eB and d e f i n e l i n e a r mappings Fn: L("X,Y)+L(X,Y)

by means o f

THE L I E GROUP OF AUTOMORPHISMS

x1,..,xneX

for GsL("X,Y),

and k = 1 , 2 , . . , n .

99

L e t us w r i t e

F=: (F1+F2+..+Fn)-F0 S i n c e 6 s a t i s f i e s ( 6 . 2 1 ) w e h a v e F ( f ) = 0 and FO,F 1 I

-

*

IFn

commute. T h e r e f o r e

.

( e x p F 1 ) . ( e x p Fn ) f = e x p ( F 1 + . . + F n ) f = ( e x p F o ) ( e x p F , ) f = ( e x p F0 ) f = =

(exp6)mf

which shows t h a t exp6cH. S i n c e B i s a l i n e a r s p a c e , w e c a n d o t h e same a r g u m e n t w i t h t 6 i n s t e a d of 6 , whence w e c o n c l u d e t h a t expt6sH f o r a l l tc3R.

# EXAMPLES. L e t A b e a Banach a l g e b r a o v e r IK ( w h e r e

6.43.

A may h a v e no u n i t a n d f a i l t o b e a s s o c i a t i v e , f o r e x a m p l e , a n y

Banach-Lie

a l g e b r a o f a n y B a n a c h - J o r d a n a l g e b r a ) . Then w e c a n

a p p l y p r o p o s i t i o n 6-42 2

t o t h e case i n which n = 2 , m= 1 , X = Y = A

i s t h e m u l t i p l i c a t i o n on A , i . e . ,

and f c L ( X , X )

f ( x , y ) = x.y.

O b s e r v e t h a t t h e a u x i l i a r y Banach a l g e b r a L ( X ) a p p e a r i n g i n p r o p o s i t i o n 6 . 4 2 i s now L ( A ) which i s a s s o c i a t i v e a n d h a s u n i t e v e n i f A f a i l s t o b e so. T h u s , t h e o r e m 6 . 4 0 a n d p r o p o s i t i o n 6 . 4 2 a r e a p p l i c a b l e . Accordingly, t h e set

w h i c h i s t h e g r o u p of a u t o m o r p h i s m s of A , i s a Banach-Lie g r o u p i n t h e normed t o p o l o g y o f L ( A )

.

The Banach-Lie a l g e b r a of

t h i s group i s

which i s t h e a l g e b r a o f l K - d e r i v a t i o n s M-algebraic equations p

of A. Besides, H i s t h e

s u b g r o u p o f G L ( L ( A ) 3 ) of d e g r e e $ 2 d e f i n e d b y t h e XPY

(a)= 0 where

100

CHAPTER

6

and x,yeA. L e t X b e a complex H i l b e r t s p a c e a n d Y = 5

. Then

can apply

p r o p o s i t i o n 6 . 4 2 w i t h n = 2 , m= 0 t o t h e spaces X a n d Y a n d t h e 2

r e a l b i l i n e a r mapping f c L ( X , E ) X,

f ( x , y ) = ( x l y ) . A c c o r d i n g l y , i f a* d e n o t e s t h e a d j o i n t o f t h e

operator H=:

g i v e n by t h e s c a l a r p r o d u c t on

C X E( X~)

,

t h e set

{ ~ c G L ( ~ ( ,E); x )

(axlay)= -

which i s t h e u n i t a r a y g r o u p of H , i s a r e a l Banach-Lie Banach-Lie

g r o u p whose

algebra is ( 6 X l y ) + ( x l d y ) =0

B= { 6 C L E ( X ) ;

The u n i t a r y g r o u p o f X is a r e a l a l g e b r a i c s u b g r o u p of G L ( L c ( X ) ,lK)

of d e g r e e 6 2 d e f i n e d by t h e e q u a t i o n s p

XIY

( a )= 0

where PX,Y

(a)= ( a x l a y ) - ( x l y )

a n d x,ysX. Let

x

b e a H i l b e r t s p a c e over 6: a n d d e n o t e by Q a n y c o n j u g a -

t i o n on X . Take f e L ( 2 X l C ) t o b e t h e r e a l b i l i n e a r mapping g i v e n by f ( x , y ) = : ( Q x / y ), x , y c X . Then w e c a n a p p l y p r o p o s i t i o n 6 . 4 2 w i t h n = 2 , m= 0 . If a

t

d e n o t e s t h e t r a n s p o s e d of t h e o p e r a t o r

ac,!&(X) , t h e n , t h e s e t

which is t h e o r t h o g o n a l g r o u p of X , i s a r e a l Banach-Lie whose Banach-Lie B = {6CLc(X);

group

algebra is

( Q 6 x / y ) t ( Q X l f i y ) =0

YX,y€X)= { 6 C L E ( X ) ; 6 t 6 t = 0 )

The o r t h o g o n a l g r o u p of X i s a r e a l a l g e b r a i c s u b g r o u p of

THE L I E GROUP O F AUTOMORPHISMS

GL(L (X)

LT

,a)

of d e g r e e c 2 d e f i n e d by t h e e q u a t i o n s p

101

X,Y

(a)= 0

where

and x,ysX.

6.44. EXERCISES. L e t X,Y b e H i l b e r t s p a c e s o v e r (I. i s a c l o s e d complex subspace &of L L T ( X , Y ) such

A J*-algebra

t h a t w e have -*As acGL(L

2 whenever A , B s

a. A

J*-automorphism i s any

,(I) such t h a t w e have LT ( A )

for a l l A , B s A .

Prave t h a t t h e group of a l l J*-automorphisms

of '2 i s a r e a l a l g e b r a i c subgroup o f G L ( L L T ( & ) , C ) of d e g r e e

c 3 . Thus i t i s a r e a l Banach-Lie group. Prove t h a t i t s BanachL i e algebra i s the set

of a l l J * - d e r i v a t i o n s of L a ( h ) .

Now w e t u r n o u r a t t e n t i o n t o t h e c o n s t r u c t i o n of a r e a l BanachL i e group s t r u c t u r e on AutD.

m o u g h o u t t h i s s e c t i o n , B c D and 6 > 0 s t a n d f o r a f i x e d open b a l l and a r e a l number such t h a t B 6 c c D . W e know t h a t (autD,

/I .I(

)

B6

i s a r e a l Banach-Lie a l g e b r a . W e d e n o t e by Holm(B,E) t h e complex Banach s p a c e of holomorphic mappings f : B+E t h a t a r e bounded on B . Thus ( H o l m ( B , E ) ,

11

./IB)

i s a r e a l Banach s p a c e ,

too *

As autD and Holm(B,E) w i l l always be endowed w i t h t h e t o p o l o g i e s

.

.I I B ,

r e s p e c t i v e l y d e f i n e d on them by t h e norms I / /I and I ] B6 w e s h a l l omit any r e f e r e n c e t o t h e s e norms. However, we s h a l l c o n s i d e r s e v e r a l t o p o l o g i e s on AutD H o l m ( B , E ) ; t h u s , i n o r d e r t o a v o i d any p o s s i b l e confusicm,

whenever w e r e f e r t o A u t D w e

102

6

CHAPTER

s h a l l e x p l i c i t e l y m e n t i o n t h e t o p o l o g y w e are c o n s i d e r i n g on i t 6.45.

LEMMA. li'herc i s

neighbourhood

R

i n a u t D s u c h t h a t , f o r any A c M

,

M

of Lhe o r i g i n

the series

13 AnidD

-1; n=O

is c o n v e r g e n t t o (expA) i n t h e s p a c e Holw(BrE). T h e m a p p i n g IB HolB(B,E) g i v e n b y

exp: M

+

(6 . 2 2 )

P r o o f : L e t AeautD b e g i v e n . Then, f o r tCiR a n d ndN, w e

have

t" An A i d e H o l m ( B r E ) ;t h e r e f o r e , w e D n!

can d e f i n e a formal

power s e r i e s IR+Holm(B,E) by means o f

t-t

(6-23)

2;

n t"! i n i d D

n =O

w e have

A s i n t h e p r o o f of p r o p o s i t i o n 4 . 1 ,

f o r a l l ndN, where M = :

/I

M =:

i d D / I B< m i s i n d e p e n d e n t of n . N o w , {AcautD;

11

All,

s 6

i s a n e i g h b o u r h o o d of t h e o r i g i n i n a u t D a n d , f o r a n y f i x e d AE

M

,

t h e r a d i u s of c o n v e r g e n c e o f

( 6 . 2 3 ) is g r e a t e r t h a n 1 .

S i n c e Hol-(B,E) i s c o m p l e t e , m

f(t,A) lB=:

C n=O

i s convergent i n Holm(BrE) t o (exptA) Moreover, i t is e a s y t o see t h a t tn-l

tn

A

n! A n i d B IB

f o r all t c [ - l , + I ] .

f ( t , A ) = C -i n i d D = A[f ( t , a ) ] dt n=l ( n - l ) !

d

THE L I E GROUP OF AUTOMORPHISMS

103

and f ( O , A ) = i d D . T h u s , b y d e f i n i t i o n 4 . 4 ,

Now ( 6 . 2 2 ) d e f i n e s a f o r m a l power s e r i e s b e t w e e n t h e r e a l Ba-

t h i s series

n a c h s p a c e s a u t D a n d H o l m ( B I E ) . S i n c e f o r AEM

i s convergent, exp: M + H o l m ( B I E ) d e f i n e s a real a n l a y t i c mapping on M

. #

6.46.

REMARK. N o t i c e t h a t e x p : M + H o l m ( B , E ) t a k e s i t s

v a l u e s n o t o n l y i n t h e s p a c e H o l m ( B , E ) b u t i n t h e smaller s e t AutD. 6.47.

LEMMA. T h e r e a r e a n e i g h b o u r h o o d

and a n e i g h b o u r h o o d expM

+

N

N

of

idD i n Holm(B,E)

of 0 i n a u t D

M

such t h a t

i s a b i j e c t i o n . Moreover, b o t h exp: M

i n v e r s e log: N

-t

a r e Z i p s c h i t z i a n on M

M

P r o o f : By lemma 6 . 4 5 e x p : M

N

and i t s

.

Holm(BIE) i s a real

+

a n a l y t i c mapping on a n e i g h b o u r h o o d

and

N

+

M of 0 i n a u t D . I t s

d e r i v a t i v e a t t h e o r i g i n i s t h e e l e m e n t of L ( a u t D , H o l m ( B , E ) ) g i v e n by e x p h l A = f; i d D = A T h u s , by t h e i n v e r s e mapping t h e o r e m , t h e r e a r e a n i e g h b o u r h o o d

M ' o f 0 i n autD and a neighbourhood

M'

such t h a t exp:

+

N'

N'

o f i d i n Holm(B,E)

i s a b i a n a l y t i c mapping. By t h e

c o n t i n u i t y of t h e d e r i v a t i v e a t 0 , t h e r e i s a convex neighbourhood

Then, f o r A1,A2e

M

o f 0 i n a u t D i n which e x p '

MI'

w e have

i s bounded

104

CHAPTER

and e x p i s l i p s c h i t z i a n o n M " .

6

A similar a r g u m e n t a p p l i e s t o

i t s i n v e r s e l o g . T h e r e i s no loss of g e n e r a l i t y i n a s s u m i n g

t h a t MI'= M

and N = expM

.

Is N = : expM a T-neighbourhood o f i d

6.48.

QUESTION.

6.49.

EXERCISE. Show t h a t i f q u e s t i o n 6 . 4 8 h a s a n

D

in

AutD?.

a f f i r m a t i v e a n s w e r , t h e n by l e m m a 6 . 4 7 , e v e r y FcAutD a d m i t s a n e i g h b o u r h o o d t h a t i s homeomorphic t o M by A+F expA. However, a s w e s h a l l see i n c h a p t e r 8 , t h e a n s w e r i s n o t a l w a y s a f f i r m a t i v e . 'Thus, i n g e n e r a l , w e may o n l y e x p e c t t h a t f o r some g r o u p t o p o l o g y , w h i c h i s f i n e r t h a n T , t h e m a p p i n g s A-+expA, AcM

, are

l o c a l homeomorphisms o f AutD o v e r a u t D f o r a l l FcAutD.

To e s t a b l i s h t h e e x i s t e n c e o f s u c h t o p o l o g y w e s h o u l d know t h a t t h e c o m p o s i t e mapping expAloexpA2 c a n a l w a y s b e w r i t t e n i n t h e form e x p C f o r some CeautD, whenever A l l A2 a r e S u f f i c i e n t l y n e a r t o 0 i n a u t D . T h i s f a c t i s a s p e c i a l case of o n e of t h e t h e main g o a l s of t h e g e n e r a l L i e t h e o r y , known a s t h e CampbellH a u s d o r f f t h e o r e m ( c f . I3 I ) 6.50.

.

T h e r e a r e a n e i g h b o u r h o o d M of t h e o r i g i n

THEOREM.

-i.n a u t D a n d a r e a l a n a l y t i c m a p p i n g C : M+autD s u e h t h a i w e h a v e

f o r a l l A1,A2eM. By t h e c o n t i n u i t y of C a t t h e o r i g i n , w e c a n f i n d neighbourhccds M l c M and M Z C M of 0 i n a u t D s u c h t h a t

C(M1, M ~ ) C M and

(6.25)

6.51.

C(M2,M2) C M 1

REMARK. The e x p l i c i t f o r m of t h e mapping C i s a l s o

known. One c a n show ( c f . C(slA1,

13 1 ) t h a t g i v e n E c a D , w e h a v e

s2A2]lB= C ( s l A 1 , =

,.

s2A2 ) i dB=

l o g ( e x p s l i l , e x p s 2 5 2 ) id,

THE LIE GROUP OF AUTOMORPHISMS

105

in the sense that the formal power series

L k,R 2 0

s:

si X k t R ( A 1,A21 %.log [id+ (expslAl exps2i2-idl]

,

where Xk, (A1,A2)= : k+1

c

=:

(-1)

n+ 1

1

c

pl+..+pn=k, ql+..+qn=R pl!ql!

n=l

piqi30

1 .. pn!q,!

-p1 3 2 A1 A2

.-

Pi +qi>o

I

"pnAqn

..A1 A2 converges in the norm

11 . I I B

to id

B

whenever

11

slAl11

and

11 s 2 A 2 I l are sufficiently small. (This is not consequence of any majorization!). Then, we necessarily have

where

and the convergence is meant in the topology of autD. Since (A1,A2)eautD for all k,R because they are partial derivatiC k,k ves in the T-sense of the mapping ( s l l ~ 2 ) + C ( s l A 1 1 ~ 2at A 2 )0, Dynkin's identity yields

k+l =

c

n=l

n+l

(-1)

1

Pn - l A ' n - lAPnAqn - 1A P I 'I . .. 1!qn! A1#A2#.-A1# 2# 1# 2

P 1!ql 1

Pn

It would be interesting to have a direct proof for the formula expAl.expA2= exp [ C Ck, (A1,AZ)] kit

in the setting of AutD.

THEOREM.I'?icr~c exists a un-iquc H a u s d o r f f

6.52.

T

6

CHAPTER

106

o n AutD such that (AutD, T a )

topology

-is a topological g r o u p a n d

1 M ; n = 1,2,..} {exp -

n

is n f u n d a m e n t a l . s y s t e m o $ n e i g h b o u r h o o o d s of i d MOreQUcr, 1'

D

for T

a

.

22'.

P r o o f : From t h e g e n e r a l t h e o r y o f t o p o l o g i c a l g r o u p s , it s u f f i c e s t o p r o v e t h a t t h e s t a t e m e n t s ( a ) , (b), ( c ) a n d ( d ) below a r e s a t i s f i e d . m

1

n

exp M= {idgl. n=1 I n d e e d , l e t fCeXpM b e s u c h t h a t f f i d D . Then, t h e r e i s some AcM

(a) W e h a v e

w i t h A 4 0 f o r w h i c h expA= f ; t h e r e f o r e , w e c a n f i n d some nEJN s u c h t h a t At#

S i n c e t h e e x p o n e n t i a l mapping i s i n j e c t i v e on

M,

M , w e h a v e ft#exp

1 n

a,

thus

A!;

f4 n

n=1

exp

n1 M .

( b ) L e t n l a n d n ClN be g i v e n ; t h e n t h e r e e x i s t s some 1 2 1 1

mdN such t h a t exp

fii

M ) fl ( e x p

M c (exp

M)

2

1

I n d e e d , i t s u f f i c e s t o c o n s i d e r m=:

.

max(nl,n2).

( c ) L e t ndN b e g i v e n ; t h e n t h e r e e x i s t s some mdN s u c h t h a t (exp

1 n

M ) . (exp

I n d e e d , by ( 6 . 2 4 ) g i v e n ndN assume M

1 ; M) ' c e x p

1 M.

we have C ( O , O ) = 0 . A s C i s c o n t i n u o u s a t 0,

we can f i n d

mm

such t h a t C (

t o b e s y m m e t r i c , i . e . , M = -M (exp

iii1

M) ( e x p

1 M) - '=

1 E

MI

. Then

1 ; M ) c 1g

M.

We may

-1 M ) = ( e x p ;I;; M I . ( e x p -

m

( d ) L e t geAutD a n d ndN b e given; t h e n t h e r e e x i s t s some mdN such t h a t g . (exp ; 1 M ) .g-'Cexp 1 M.

I n d e e d , once gcAutD h a s b e e n f i x e d , by c o r o l l a r y 5 . 1 2 t h e a d j o i n t mapping gy':

autD+autD o f g - '

i s a n a u t o m o r p h i s m of t h e

107

THE L I E GROUP OF AUTOMORPHISMS

-1

a l g e b r a autD. T h e r e f o r e t h e s e t g # (

Banach-Lie

1 n

M)

is a

neiqhbourhood of 0 i n autD and w e may f i n d some m a such t h a t 1 m

-1

M c q #

so t h a t

1

(

M). Moreover, by p r o p o s i t i o n 5 . 1 3 ,

g.exp(

1

M)q

-1

cexp

I n

M.

I n o r d e r t o show t h a t T >,T it s u f f i c e s t o prove t h a t e v e r y T-neighbourhood of i d D c o n t a i n s a T -neighbourhood of i d

D

.

Now,

t h e f a m i l y of s u b s e t s o f AutD g i v e n by

f o r E > O i s a fundamental system of T-neighbourhoods of i d D . By

lemma 6 . 4 5 t h e mapping exp: M+Holm(B.E) i s c o n t i n u o u s a t t h e o r i g i n ; as

i s a neighbourhood of i d D f o r t h e t o p o l o g y induced by H m ( B , E ) on AutD, t h e r e e x i s t s some ndN such t h a t e x p ( 1 M ) c N ( E ) .

# 6.53.

DEFINITION.

We r e f e r t o t h e t o p o l o g y i n t r o d u c e d by

t h e o r e m 6 . 5 2 on A u t D a s t h e “ a n a Z y t i c t o p o l o g y “ o n AutD. By A u t O D w e d e n o t e t h e connected component o f i d D i n (AutD, T a ) .

6.54.

LEMMA. T h e r e i s a n e i g h b o u r h o o d M o f t h e o r i g i n i n

autD s u c h t h a t exp: M-texpM

i s a homeomorphism when b o t h M and

a r e endowed w i t h t h e i r r e s p e c t i v e t o p o l o g i e s a s s u b s p a c e s of autD and ( A u t D , T). expM

P r o o f : L e t M be as i n theorem 6 . 5 0 .

By lemma 6 . 4 7 ,

exp: M-+Holm(BIE)i s a homeomorphism of M o n t o a neighbourhood

expM of i d D i n H o l m ( B I E ) . Now it s u f f i c e s t o r e a l i z e t h a t exp t a k e s i t s v a l u e s n o t o n l y i n Holm(BIE) b u t i n t h e s u b s e t AutDcHol,(B,E)

and t h a t t h e t o p o l o g y induced by Holm(B,E) on

AutD i s p r e c i s e l y T .

ff

CHAPTER

108

6.55.

6

REMARK. Observe t h a t e x p : M+expM 1s a homeonmrphism,

t o o , f o r t h e t o p o l o g i e s i n d u c e d on M

a n d expM

by autD and

(AutD, T,). Thus, i n p a r t i c u l a r , T and 4 a g r e e on t h e s u b s e t expM o f AutD, b u t from t h i s f a c t w e c a n n o t c o n c l u d e t h a t T a n d I' a g r e e on t h e whole g r o u p AutD: w h e r e a s expM of i d D f o r T I it may f a i l t o be so f o r T . 55.-

i s aneighbourhood

The Banach-Lie o up s t r u c t u r e of AutD. ____ - - - -___-g r-

Now w e are g o i n g t o c o n s t r u c t a r e a l Banach-Lie g r o u p s t r u c t u r e on AutD whose u n d e r l y i n g t o p o l o g y is T For t h i s p m p o s e , l e t and M 2 b e a s i n t h e o r e m 6 . 5 0 and 6 . 5 2 so t h a t M ,M1

.

C ( M 1 , M I1

(6.26)

=M

C ( M 2 , M 2 )= M I

and exp: M-bexpM i s a homeomorphism f o r t h e t o p o l o g i e s i n d u c e d by a u t D and (AutD, T,). L e t u s d e n o t e by l o g : expM-tM i n v e r s e and w r i t e F=: I N ; N

its

open and O e i l r c M ]

Then, t h e f a m i l y IexpN; NcF} i s a f u n d a m e n t a l s y s t e m of n e i g h b o u r h o o d s o f i d D f o r T,. 6.56.

THEOREM. T h e r e is a u n i q u e r e a l a n a l g t i c B a n a c h

r n a n i f o Z d s t r u c t u r e on (AutD, T ) f o r w h Z c h t h e f a m i l y a

is a s y s t e m o f c h a r t s a t t h e i d e n t - i t y e l e m e n t id D' Proof:

L e t geAutD be g i v e n . The l e f t t r a n s l a t i o n

Lg: f + g o f , fcAutD, is a n automorphism o f t h e t o p o l o g i c a l -1 Now w e d e f i n e g r o u p (AutD, Ta) whose i n v e r s e i s ( L g ) - ' = Lg a s y s t e m of c h a r t s a t geAutD a s t h e family of p a i r s

.

C l e a r l y , c o n d i t i o n M1 of d e f i n i t i o n 6 . 1 is s a t i s f i e d b e c a u s e

THE L I E GROUP O F AUTOMORPHISMS

U

109

g.expN

NeF, g e A u t D i s an open c o v e r of A u t D f o r t h e t o p o l o g y Ta. Moreover, t h e s e

l o c a l c h a r t s are a n a l y t i c a l l y c o m p a t i b l e i n t h e r e a l s e n s e ,

i.e.,

t h e y s a t i s f y c o n d i t i o n M 2 , t o o . Indeed, assume t h a t

f o r some g ,g2eAutD and N 1 , N 2 c F . 1

AleN

1

Then, t h e r e are A l c N l

and

such t h a t gl.expA = f = g2.expA2 1

w e have

Thus, f o r AcNl n N 2

=

S i n c e A1,A2cM2

,

log [expAlexp ( -A2) expA]

from theorem 6.50 w e d e r i v e

expAlexp(-A2) = expC(A1 , - A 2 ) = expA 3 where, by ( 6 . 2 6 )

, A 3 =:

C ( A ,-A2) 1

i s a f i x e d e l e m e n t of M

1'

Then,

from theorem 6.50 w e d e r i v e l o g [expA exp (-A 1

2

) expA] = l o g (expA expA) = 3

Whence t h e t r a n s i t i o n homeomorphism c o r r e s p o n d i n g t o t h e c h a r t s (glexPN1 I

and (g2expN2' 1oglg2expN2

i s g i v e n by

A J C ( A ~ , A ) which i s a r e a l a n a l y t i c mapping.

# 6 . 5 7 . THEOREM. The m a n i f o l d

(AutD, T a ) is a r e a l Banach-

L i e g r o u p whose B anach- Li e a l g e b r a i s autD.

110

CHAPTER

6

P r o o f : By p r o p o s i t i o n 6 . 1 9 a n d 6 . 3 0 it s u f f i c e s t o p r o v e t h e s t a t e m e n t s ( a ) , (b) and ( c ) below. ( a ) F o r e v e r y f i x e d gcAutD, t h e mapping Lg: q+gf i s r e a l a n a l y t i c on AutD. I n d e e d , l e t gcAutD b e f i x e d . Choose a n y fcAutD 1

c h a r t ( f .expN, l o g L f ( g f e x p u , l o g L (gf)

and a n y l o c a l

1 of (AutD, T a ) a t f . Then

I gfexpN

i s a l o c a l c h a r t a t g f and t h e

e x p r e s s i o n o f Lg i s t h e s e c h a r t s i s g i v e n by

= l o g expA= A

for A e N . Thus, Lg i s a n a l y t i c . ( b ) For e v e r y f i x e d gcAutD, t h e mapping

Tg: AutD+AutD g i v e n by f + g f g - l i s r e a l a n a l y t i c . I n d e e d , l e t gcAutD be g i v e n . Choose any fcAutD

and a n y l o c a l

1

c h a r t (f.expN, logLf-

) a t f . Since t h e a d j o i n t !f.expN g # : autD+autD o f g i s c o n t i n u o u s , t h e r e i s some N C M 1

t h a t g# (iV,)CrV; t h e r e f o r e ,

mapping such

l o g e x p g # ( A )= g#A

€ o r A c N l . Then ( f . e x p N 1 , l o g L f - l l f e x p N ) a n d (gfg-lexPNl' logL(gf9

-1

1

-1

1

1 are local c h a r t s a t f

Igfg-lexpN 1

and g f g - l . Moreover, t h e e x p r e s s i o n of Tg i n t h e s e c h a r t s i s g i v e n by

for a&N

1'

Thus T g i s a n a l y t i c .

( c ) The mapping F:

(AutD)x(AutD)+AutD g i v e n by

111

THE LIE GROUP OF AUTOMORPHISMS

(f,g)+fg-' is analytic in a neighbourhood of ( idD,idD)

.

Indeed, let M2 be as in (6.26). Then (expM2, loglexpM21 is a chart of AutD at idD and its "Cartesian square" is a chart of AutDxAutD at (idD,idD) By (6.25) we have

.

for all A1,A2eM2. Then, it is easy to check that the expression of F in these charts is given by (A1'A2 which is real analytic. 56.- The action of AutD on the domain D. ~~

We endow the domain D with its underlying real analytic manifold structure and consider DxAutD as a product manifold. Then we define the action of AutD on D as the mapping JI: DxAutD+D given by (x,f)+f (x)

.

6.58. THEOREM. T h e mapping tic o n DxAutD.

+:

(x,f)+f(x) is r e a l a n a l y -

Proof: It suffices to prove its analyticity near the identity element idD. Now, let xcD be given and fix any ball B c c D centered at x. Starting with this ball B we can construct a neighbourhood M of 0 in autD as we did in 54. Then (B, idB)x(expM, log,expM)is a local chart of D AutD at (x,idD). Also, (D, idD ) is a chart of D at x. Thus, it suffices to show that the mapping

which is the expression of JI in these charts, is real analytic in BxM. By lemma 6.45, (y,A)+(y,expA) is real analytic in BxM with

112

CHAPTER

6

v a l u e s in B x H o l m ( B , E ) . O b v i o u s l y , t h e mapping B x H o ~ ~ ( B , +E E ) g i v e n by ( y , f 1 +f (y) is s e p a r a t e l y h o l o m o r p h i c ;

t h e r e f o r e , by H a r t o g ' s t h e o r e m , it is h o l o m o r p h i c a n d , i n p a r t i c u l a r , real a n a l y t i c . But ( 6 . 2 7 ) i s t h e c o m p o s i t e of ( y , A ) + ( y , e x p A ) and (y,f)+f ( y ) , whence t h e r e s u l t follows.

#

CHAPTER

7

BOUNDED CIRCULAR DOMAINS

I n t h i s c h a p t e r w e s h a l l s t u d y t h e group A u t D f o r domains D w i t h some p a r t i c u l a r g e o m e t r i c p r o p e r t i e s .

§I

.- The

L i e a l g e b r a autD f o r c i r c u l a r domains.

7.1.

DEFINITION. We s a y t h a t a b o u n d e d d o m a i n D

is

“ c i r c u l a r ” i f OeD a n d , f o r a l l xcD a n d a l l AcC w i t h I A / = 1

,

we h a v e AxeD. Throughout t h e whole c h a p t e r , D w i l l s t a n d f o r a bounded c i r c u l a r domain.

7 . 2 . LEMMA. L e t D b e a b o u n d e d c i r c u l a r d o m a i n . T h e n t h e v e c t o r f i e l d Z : x + i x i s c o m p l e t e i n D.

tm,

it

x. S i n c e D i s c i r c u l a r , w e have f c A u t D and t h e mapping t + f t i s a T-continuous one-parameter group. By theorem 4 . 5 i t s a s s o c i a t e d v e c t o r f i e l d , which i s o b v i o u s l y 2, i s complete i n D. P r o o f : For

w e d e f i n e f t : x+e

t

# W e c a l l Z t h e c i r c u l a r v e c t o r f i e l d and it w i l l p l a y an i m p o r t a n t r o l e i n t h e s t u d y of c i r c u l a r domains. S i n c e O c D , any (non n e c e s s a r i l y c o m p l e t e ) holomorphic v e c t o r f i e l d X i n D i s u n i q u e l y determined by i t s T a y l o r s e r i e s a t 0 . W e w r i t e Pn f o r t h e space of c o n t i n u o u s n-homogeneous CU

polynomials P: E+E, so t h a t w e have X= C P = : X ( nS P O

n=O n

for nm.

113

P n where

CHAPTER

114

z#=: [ Z , . ] be t h e a d j o i n t of 2 ; and , by r e i t e r a t i n g t h i s o p e r a t i o n

Let

7

t h e n w e may a p p l y Z #

to X

a n d t a k i n g l i n e a r combinations,

w e o b t a i n e x p r e s s i o n s of t h e form P(z#)x= (ao+a

z + . . + ar Z '#) X

1 #

where P ( A ) = a o + a l A t . . + a r A r

i s a polynomial i n t h e indetermi-

nate A .

1.3. LEMMA. LeL P(X)c(I:[A] b e a n y poZynumiaZ i n A a n d m wssume t h a l X = l' P ,is a h o l - o m o r p h i c v e c t o r J i e l d i n D. T h e n n=O n we have

P r o o f : F o r t h e homogeneous components P

of X w e h a v e

By r e i t e r a t i n g t h i s o p e r a t i o n a n d t a k i n g l i n e a r c o m b i n a t i o n s

we o b t a i n t h e r e s u l t .

# m

7.4.

LEMMA.

Assume t h a t X = 1 P n s a t i s f i e : : XeautD. Then

we h a v e P = 0 f o r a l l 1-113.

n=O

n

P r o o f : L e t XeautD b e g i v e n . S i n c e a u t D i s a r e a l L i e a l g e b r a and ZcautD, w e h a v e P ( Z # ) X s a u t D f o r a n y p o l y n o m i a l w i t h r e a l c o e f f i c i e n t s P(A)dR[A].

By t a k i n g P ( A ) = A 3 t A

and

a p p l y i n g lemma 7 . 3 w e o b t a i n

B u t now we h a v e P ( - i ) = P ( 0 ) = P (i) = 0 , so t h a t t h e T a y l o r series of P ( Z # ) X a t 0 i s

BOUNDED CIRCULAR DOMAINS

f o r k= 0 , l a n d , by C a r t a n ' s

[P(Z#)X] (k= 0

Thus, we have

115

0

u n i q u e n e s s theorem, P ( Z # ) X = 0 . However, P ( n i - i ) f O f o r a l l n > 3 ; therefore P = 0 for n>3. DEFINITION. For a n y b o u n d e d c i r c u l a r d o m a i n D , we

7.5.

set 0

a u t D =:

P1

n

aut D=:

(autD)

0

0

(autD)O= { X ( o ) ; XeautDj

E =:

Aut D = :

(POOP2)n autD

{FsAutD, F i s l i n e a r ] .

PROPOSITION. F o r b o u n d e d c i r c u l a r d o m a i n s D ,

7.6.

we

h a v e t h e t o p o Z o g i c a 2 d i r e c t sum d e c o m p o s i t i o n 0

autD= ( a u t o D )8 ( a u t D)

(7.1)

M o r e o v e r , a u t 0 D i s t o p o l o g i c a 2 2 y i s o m o r p h i c w i t h E0 ( c o n s i d e r e d 0 a s a r e a l l i n e a r s u b s p a c e of E l b y t h e m a p p i n g X - t X ( 0 ) a n d a u t D 0

c a n v i e w e d a s t h e L i e a Z g e b r a of Aut D. P r o o f : L e t XcautD be given: by lemma 7 . 4 w e have f o r some P k e P k , k = 0,1, 2 .

X = P +P + P 0

1

2

Applying lemma 7 . 3 t o

t h e polynomial P (1)= h 2 and t h e v e c t o r f i e l d X w e d e r i v e P(Z

#

)x=

2

2

2

1 P(ni-i)P = C (in-i) P = -(Po+P2) n n

n=O

n=O

so t h a t P0+PZ€autDa n d , t h e r e f o r e , P 1= X - ( P 0 + P 2 ) c a u t D . C l e a r l y 0

P + P 2 c a u t D and P l c a u t D ; t h u s autD a d m i t s t h e d i r e c t sum 0 0 0 decomposition autD= ( a u t D ) Q ( a u t o D ) By lemma 5.1 I , t h e c a n o n i -

.

2

c a l p r o j e c t o r s Z#

and

I-Z#

2

are c o n t i n u o u s .

Now, l e t c s E O be g i v e n . Then, t h e r e e x i s t s a unique symmetric b i l i n e a r mapping QccL( E x E I E ) such t h a t t h e v e c t o r f i e l d A:

x + c - Q c ( x , x ) , XCD, b e l o n g s t o autD. I n d e e d , t h e r e i s some

X = P + P +P cautD w i t h c = X ( 0 ) = P o . Then w e have c+P2cautD and 0

1 2

,

requirements. I f t h e r e i s

Q ( x , x )=: -P2

(XI

a n o t h e r Q:

i n t h e same c o n d i t i o n s , from c-QccautD and

I

xcD

c-QA cautD w e g e t Qc-Q:=

satisfies the (c-QA

)-(c-Qc)cautD; t h u s

CHAPTER

116

f o r k= 0 , l

(Qc-QA)Ak= 0

Q,=

7

and, by C a r t a n ’ s u n i q u e n e s s theorem,

QL-

Now, w e show t h a t Eo i s complete and t h a t t h e mapping

E o + a u t D g i v e n by c+c-Q i s a c o n t i n u o u s s u r j e c t i v e i s o m r p h i s m 0 of Banach s p a c e s . Indeed, assume t h a t Q = 0 f o r some c c E A s c-QccautoD, w e have ccautD. From ZeautD we o b t a i n

0’

[ Z , c ] = i c s a u t D . S i n c e autD i s p u r e l y r e a l , w e have c = 0 . Thus,

i s a n isomorphism o n t o t h e image subspace which i s o b v i o u s l y a u t D. NOW, l e t u s t a k e any b a l l B c c D c e n t e r e d a t 0 OeD. By theorem 5.6, t h e r e a r e c o n s t a n t s K 1 , K Z such t h a t w e CW-Q

have

f o r a l l XeautD. Applying t h i s t o t h e v e c t o r f i e l d X = c-Q e a u t D w e o b t a i n 0

for a l l c e E o . Thus c+c-Q

i s a homeomorphism. S i n c e w e know t h a t a u t D i s c l o s e d i n a u t D , Eo i s complete. A s f o r t h e

0 a s s e r t i o n c o n c e r n i n g a u t 0D , w e c a n r e p e a t t h e arguments of

theorems 6.56 and 6.57 r e s t r i c t i n g o u r s e l v e s t o t h e group 0 A u t D i n s t e a d of AutD.

I n t h e c o u r s e of t h e proof w e have e s t a b l i s h e d t h e f o l l o w i n g 7.7.

Eo i s a r e a l s u b s p a c e of E a n d , f o r e a c h a unCque QceL(EXEIE) s u c h t h a t t h e v e c t o r f i e l d

COROLLARY.

c e E O , t h e r e is

x*c-QC(xIx), XCD, b e l o n g s t o a u t D .

7.8.

DEFINITION. Ve r e s e r v e t h e n o t a t i o n Qc f o r t h e

symmetric biZCnear mapping d e s c r i b e d above. 7.9.

PROPOSITION.

For bounded c i r c u l a r dom ai ns D ,

h a ve 0

0

0

[aut D , a u t D] c a u t D ,

0

[aut D , a u t 0 D ] c a u t O D

we

BOUNDED CIRCULAR DOMAINS

0

117

0

[ a u t D , autoD] c a u t D

Mor.eover, 0

( a ) For a l l L c a u t D , c c E o a n d x c E , it holds L c c E O a n d

Q L C ( x , x ) = LQ ( x , x ) - ~ cQ( L x x) 0

( b ) For a l l c 1 , c 2 e E 0 , We h a v e Q

( . , c 2 ) + Q c( c l , . ) c a u t D 2

,1

( c ) For xcE a n d c 1 ,c2eE0 t h e following e q u a l i t y h o l d s

[Q,

Q, 1

( x , x ) , X I = Qc [Qc ( X , x ) , X I . 2

2

1

0 ( d ) For a l l FcAut D , c c E O a n d XEIE, we h a v e FccEO a n d QFc ( x , x ) = FQc ( F - l x ,

F-lx)

.

0

Proof: L e t L , L 2 c a u t D be 9 i v e n ; t h e n

so t h a t Banach-Lie

0

[aut D,

1

0 a u t o D ] c a u t D.

[L 1 , L 2 ]

is l i n e a r

I n p a r t i c u l a r , autoD i s a

s u b a l g e b r a of a u t D .

0 L e t L s a u t D a n d AeautOD b e a r b i t r a r i l y g i v e n ; t h e n w e h a v e

A ( x ) = c - Q c ( x , x ) , X C D , where c s E O a n d Q c i s a s y m m e t r i c b i l i n e a r mapping ExE-tE. An e a s y c o m p u t a t i o n g i v e s

S i n c e t h e mapping x+2Qc(Lx,x)-L[Qc~x,x))is a n e l e m e n t of P 2 , i t $0

2 ZOWS

[L,A] cautOD. B e s i d e s L c =

[L ,A] O s E o

and

which p r o v e s ( a ) . L e t A 1 , A 2 c a u t D b e g i v e n and assume t h a t 0

where c l , c 2 e E

0

follows t h a t

,

and Qc l

e L ( E x E 1 E ) a r e symmetric.

Q C2

It

7

CHAPTER

118

S i n c e t h e mapping x+QC ( x , Q c ( x , x ) 1 -Q, 1

2

1

(Qc ( X , X ) , X I 2

,

XED,

is

a n element of P 3 , by lemma 7 . 4 , it m u s t be i d e n t i c a l l y n u l l . I C ~ +Qc ) ( c l , . ) e a u t 0 D. T h i s p r o v e s ( b ) and

Thus [A1 ,A2] = Q c ( . (c)

.

1

2 0

F i n a l l y , l e t FcAut D and acEO be g i v e n . Then w e have F = expL 0

f o r XCD, Fx i s t h e v a l u e a t t = 1 of t h e

f o r some L s a u t D , i . e . ,

s o l u t i o n of t h e i n i t i a l v a l u e problem

i n t h e space E . By ( a ) w e have L ( E O ) C E o ; t h e r e f o r e , i f t h e i n i t i a l v a l u e i s some C c E o f l D , ( 7 . 2 ) can be i n t e r p r e t e d a s an i n i t i a l v a l u e problem i n E S i n c e E i s complete and t h e 0' 0 s o l u t i o n of ( 7 . 2 ) i s u n i q u e , w e have Fc= (expL)ccE f o r a l l 0 ccEOfl D. A s F is l i n e a r , FceEO for a l l c s E o . Moreover, F#[c-Q, ( x , x ) ] = F[c-Q,

( F - l x , F - ' x ) ] = Fc-FQ, (F-'x,F-'x)

f o r a l l xeE, so t h a t

Q,,

( x , x ) = FQ,

(F-'X,F-'~)

which shows ( e l .

#J 7.10.

COROLLARY.

T h e s u b s p a c e E o is i n v a r i a n t u n d e r t h e

g r o u p A u t 0D. In p a r t i c u l a r , E

0

is a c o m p l e x s u b s p a c e of E . T h e 0

m a p p i n g c+Q is c o n j u g a t e l i n e a r a n d w e h a v e Q , ( c , . ) e a u t D f o r C

a l l ceEo. 0

P r o o f : W e have F=: i d cAut D D

because D i s c i r c u l a r .

Applying (e) we g e t Q i c = -iQc. Then a p p l y (b) w i t h

BOUNDED CIRCULAR DOMAINS

7.11.LEMMA

.

We h a v e E o = t X ( c ) ; c c E O

119

I

XcautDl.

Proof: Let u s set ( a u t0 D ) E = : { L ( c ) ; c c E o l L s a u t 0 D)

I

( a u t o D ) E o = { A ( c ) ; ccEO,AcautoD}

0

F i r s t w e show t h a t (aut$)EocEo. I n d e e d , l e t c c E O be g i v e n and

t a k e any Lcaut 0D . Then A = : c-QccautD so t h a t

a t 0 w e g e t L ( c ) c E O and t h e r e f o r e

By e v a l u a t i n g [L,A] ( a u t0D ) E C E 0

[L,A]cautD. But

0

Now w e show t h a t ( a u t o D ) E o c E o . I n d e e d , l e t c c E O be g i v e n . Then A= c-Q cautD; s i n c e ZsautD, w e have A 2# ( Z ) . = [A,[A,Z]]cautD.

[A, [A,Z]]x=

4iQ (x,c)

But

xcD

S i n c e t h e mapping x + 4 i Q c ( x , c ) i s l i n e a r , w e must have 0 [A, [A,Z]]caut D and by t h e p r e v i o u s s t e p w e g e t Qc ( x , c )cEO

(7.3)

f o r a l l c e E O and xcE t h e symmetry of Q,

0'

I n t e r c h a n g i n g t h e r o l e s of x and c , by

we g e t Qx(c,x)cE0

(7.4)

f o r a l l x,ccEO. As c+Q,

is linear,

Thus, from ( 7 . 3 ) and (7.4) w e d e r i v e Q C ( x + c , x ) c0 ~ f o r a l l x , c e E O . Then, from QC(X+C,X)=

Q C t x , x ) + Q ( c , x ) e E0

and ( 7 . 3 ) we o b t a i n Q c ( x , x ) c E o f o r a l l xcEol SO t h a t

120

7

CHAPTER

( a u t o D )Eo c E O . by p r o p o s i t i o n 7 . 6

NOW,

( a u t o )E g = : { X ( c ) ; ccEO

i t follows

, X c a u t D ] c ( a u t0 D)E o + ( a u t o D )E O c E o .

The c o n v e r s e i n c l u s i o n i s o b v i o u s .

52.-

The c-o.n-n-~~ e c t e d component of t h e i d e n t i t y i n- AutD. -

--~

L e t cT b e a n y H a u s d o r f f t o p o l o g i c a l g r o u p a n d d e n o t e by W a n

o p e n s y m m e t r i c c o n n e c t e d n e i g h b o u r h o o d of t h e i d e n t i t y e l e m e n t e i n J . F o r nCN w e s e t

From t h e g e n e r a l t h e o r y of t o p o l o g i c a l g r o u p s , it i s known that

lY= :

U

W"

nEN

i s a c l o s e d n o r m a l s u b g r o u p of J a n d t h a t H i s t h e c o n n e c t e d

component of e i n J ; t h u s , H d o e s n o t d e p e n d o n t h e c h o i c e o f W.

S i n c e t h e mapping e x p i s a l o c a l homeomorphism a t 0 , i n

p a r t i c u l a r we get. 7.12.

LEMMA. L e t M be t h e n e i g h b o u r h o o d of 0 in a u t D

g i v e n b y Zernma 6 . 5 6 .

Then

is t h e c o n n e c t e d c o m p o n e n t o f i d D i n ( A u t D , T a ) . M o r e o v e r , Aut D i s a c l o s e d norvnul szibgroup of b o t h (AutD,T ) a n d 0 a (AutD,Y). 7.13.

DEFINITION. L e t S and

autD r e s p e c t i v e l y .

G

be s u b s e t s of D and

We d e f i n e t h e " o r b i t " of S b y

I t i s immediate t o check t h a t i f

G

G

by meuns o f

i s a s u b g r o u p of AutD, t h e n

121

BOUNDED CIRCULAR DOMAINS

w e have

GG ( S ) =

G(S).

L e t E o be t h e subspace o f E g i v e n by d e f i n i t i o n 7 . 5 . C l e a r l y E

0

n D i s a bounded open c i r c u l a r s u b s e t of t h e s p a c e E

however, E o n D

-

0'

may f a i l t o be connected

LEMMA. I f D i s a bounded c i r c u l a r domain of E,

7.14.

t h e n ue have

(AutoD) ( E o n D ) c E o f l D .

P r o o f : L e t gcAut D be g i v e n . By lemma 7 . 1 2 w e have 0

g= ( e x p A l ) o .

.O

(expA ) f o r some AkcM,

k = 1 , 2 , . . , n . Thus, i t

s u f f i c e s t o show t h a t

f o r a l l AcM. NOW, l e t AeM and xeEOn D be g i v e n . L e t us c o n s i d e r t h e i n i t i a l

v a l u e problem

(7.5)

i n t h e space E . I t s s o l u t i o n y ( t ) = (exptA)x s a t i s f i e s y ( t ) e D for a l l

tm.

S i n c e xcEO and by p r o p o s i t i o n 7 . 6 Eo i s c o m p l e t e ,

t h e i n i t i a l v a l u e problem ( 7 . 5 ) h a s a s o l u t i o n i n E o , t o o . A s t h e s o l u t i o n i s u n i q u e , w e have (exptA)xcEo f o r a l l t d R ; t h u s , (expA) ( E n~ D ) c E~ n D .

# 7.15. LEMMA. L e t D be a b o u n d e d c i r c u l a r domain of E .

T h e n , i f E n D i s c o n n e c t e d (in p a r t i c u l a r , i f D i s b a l a n c e d ) , 0

we h a v e

Proof: A s E

0

nD

i s assumed t o be c o n n e c t e d , E o n D - i s a

bounded c i r c u l a r domain o f t h e s p a c e E o and it makes s e n s e t o

122

7

CHAPTER

speak

.

of t h e group A u t ( E o n D)

L e t qsAutOD be g i v e n . Then g i s a biholomorphic b i j e c t i o n of D

o n t o D; t h e r e f o r e q

i s a biholomorphic b i j e c t i o n o f

lEon D E o n D o n t o i t s image g ( E o il D)

g(E0n

(7.6)

. From

lemma 7 . 1 4 ,

DICE^^

D

Thus, a p p l y i n g gT1eAut0D t o ( .6) and lemma 7 . 1 4 a g a i n , w e obtain

~~n

E o n Dcg-'

c~-' ( E o n D) = Eo

so t h a t

n D.

D)C

~ , nD

Then g ( E 0 n D ) = E o n D and

.

eAut(E,, fl D ) fl D Observe t h a t , i f D i s b a l a n c e d , t h e n E 'lEO

0

nD

i s balanced t o o ;

hence i t i s connected and t h e lemma h o l d s .

#

7.16. LEMMA. L e t D be a boun ded c i r c u l a r domain of E .

T h e n , if Eo n D is c o n n e c t e d ( i n p a r t i c u l a r , if D is b a l a n c e d ) , the set

( A u t o D ) O i s a n e i g h b o u r h o o d of 0 i n t h e s p a c e E o .

P r o o f : Consider t h e mapping

0:

Eo+aut D g i v e n b y ~ 4 c - Q ~ . 0

i s a neiqhbourhood of 0 i n autD, M n autDo i s a neighbourhood of 0 i n a u t D and, by p r o p o s i t i o n 7 . 6 , Since M

U=:

+-'Mn

0

autoD i s a neighbourhood of 0 i n E o . NOW, c o n s i d e r

t h e composite J of t h e mappings C+C-Q

-t

exp (c-qC)

+

"

By lemma 6.32 w e have J ( c ) = C

1

[exp ( c - ~ , )] o (i:idD)O f o r ceU, where w e

n=O

have p u t A c = :

c-Qc. Moreover, we have ( i : i d D ) = 0 f o r a l l n f l

-1 and (Acid,)O= c

,

so t h a t J ( c ) = c f o r ceU and J f U ) is a neighbourhood of 0 . Thus, by lemma 7 . 1 2 . OeU= J ( U ) =

(expAc)O; csU}c{expA)O;

a n d , by lemmas 7.15 and 7 . 1 4 ,

A e M I c (AutoD)0

BOUNDED CIRCULAR DOMAINS

123

whence t h e r e s u l t f o l l o w s .

# 7.17.

PROPOSITION. L e t D b e a ( n o n n e c e s s a r i z y c i r c u Z a r l

b o u n d e d d o m a i n i n E a n d l e t J b e a s u b g r o u p o f AutD s u c h that, for some XGD, t h e orbit J ( x ) o f x b y J i s a n e i g h b o u r h o o d of x. T h e n J ( x ) = D a n d t h e s u b g r o u p J acts t r a n s i t i v e Z y o n D. P r o o f : F i r s t w e show t h a t J ( x ) i s a n open s u b s e t of D . L e t y s J ( x ) be g i v e n . Then w e have g ( x ) = y f o r some g c J . A s J ( x )

i s assumed t o be a neiqhbourhood of x , t h e r e e x i s t s some open

s u b s e t W c D such t h a t xcWCJ(x) and, a p p l y i n g g w e o b t a i n

S i n c e q i s a homeomorphism, g(W) i s open; t h u s , by t h e a r b i t r a r i n e s s of y , J ( x ) i s open. Now w e show t h a t J ( x ) k a c l o s e d s u b s e t of D. L e t ycD b e any

-

p o i n t of t h e closure J ( x ) of J ( x ) i n D . Then, t h e r e i s a sequence ( y n )n C NJC ( x ) s u c h t h a t y,*y.

T h e r e f o r e , w e have

f o r some q n CJ and ndN. L e t dD be C a r a t h e o d o r y d i s t a n c e i n D . S i n c e w e have assumed t h a t J ( x ) i s a

y,= g,(x)

neighbourhood o f x and, by c o r o l l a r y 3 . 1 4 ,

d,

i n d u c e s t h e norm

t o p o l o g y on D ,

f o r some E > O . Moreover, a s y,+y

it f o l l o w s t h a t

f o r a l l n m 0 . Since dD is J - i n v a r i a n t , dD(x,q,ly)i'

so t h a t , by ( 7 . 7 )

,

from ( 7 . 8 ) w e o b t a i n and t h e r e f o r e

0

J ( x ) = J ( x ) . Thus J ( x ) i s c l o s e d i n D .

ycg, 0

S i n c e D i s c o n n e c t e d , w e have J ( x ) = D and D i s homogeneous

CHAPTER

124

7

under the action of J .

# 7.18. COROLLARY. L e t D b e a b o u n d e d c i r c u l . a r d o m a i n of .if E o n D i s c o n n e c t e d ( i n p a r t i c u l a r , if D i s baZancedJ ~e h a v e (AutoD)O= Eon D. E.

Then,

Proof: Consider the Banach space E the bounded domain 0' 7.15, Aut D is a 0 subgroup of Aut(Eo n D) ; by lemma 7.16 (AutoD)0 is a neighbourhood of 0 in E o n D. Then, proposition 7.17 gives the result. E o n D and the point OcEOn D. By lemma

ff 5 3 . - Study of the orbit origin. ___ (AutD)O of the-___-

In order to make a deeper study of the orbit (AutD)O of the origin we recall some properties of analytic sets. 7.19. DEFINITION. A s u b s e t R of domain D i s s a i d t o b e i n D if, f o r e v e r y p o i n t xcD, t h e r e i s a n e i g h b o u r h o o d U of x a n d t h e r e is a s e t F cHol(U,!l) of h o l o m o r p h i c f u n c t i o n s f: U+E s u c h t h a t we h a v e

complex-analytic

Roughly speaking, a subset D of D is analytic in D if, and only if, R can be locally represented a s the "joint kernel" of a set of holomorphic functions. For a study of the elementary properties of analytic sets see for example 1 4 5 1 p. 50. DEFINITION. L e t D b e a b o u n d e d c i r c u l a r d o m a i n in E. We d e n o t e b y R t h e s u b s e t o f D c o n s i s t i n g of t h e p o i n t s x c D for w h i c h (AutoD)x is a c o m p l e x - a n a l y t i c c l o s e d s e t i n D. 1.20.

LEMMA. L e t D b e a b o u n d e d c i r c u l a r d o m a i n i n E . T h e n , i f Eon D i s c o n n e t e d ( t h u s , i n p a r t i c u l a r , when D i s b a l a n c e d ) , u e h a v e Ll#$ a n d (AutD)R=R. 7.21.

BOUNDED CIRCULAR DOMAINS

125

P r o o f : S i n c e E n D i s assumed t o be c o n n e c t e d , by 0

c o r o l l a r y 7.18 w e have (AutoD)O=E

0

fl D .

Since Eo i s a c l o s e d

complex subspace of E l by t h e Hanh-Banach E

s e p a r a t i o n theorem,

f l D i s a complex-analytic c l o s e d set i n D;

0 and R f $ .

t h u s se have O c R

Let x c ( a u t D ) R ; t h e n t h e r e a r e ycR and gcAutD such t h a t gy= x . S i n c e by lemma 7 . 1 2 AutOD i s a normal subgroup of AutD, w e have ( a u t o D ) x = (AutoD)gy= g ( a u t o D ) y . B u t (AutoD)y i s a complexa n a l y t i c c l o s e d set i n D and t h e s e p r o p e r t i e s a r e p r e s e r v e d by gsAutD; t h u s , xcR and ( A u t D ) R c R . The o p o s i t e i n c l u s i o n i s obvious. 7.22.

# LEMMA. L e t D b e a b o u n d e d c i r c u l a r d o m a i n f o r

w h i c h E fl D i s c o n n e c t e d . 0

T h e n , f o r a l l xcll, we h a v e

P r o o f : I f x=O, t h e n t h e a s s e r t i o n i s t r i v i a l . L e t xcR be g i v e n w i t h x#O and p u t V = :

i

{Act; I A / < I /x / I - 1. S i n c e (AutoD)x

i s a complex-analytic c l o s e d s e t i n D ,

i s a complex-analytic c i r c u l a r group f

t

c l o s e d s u b s e t of V . A s D i s c i r c u l a r , t h e

(y)=: e

it

y , tCR, ysD, i s c o n t a i n e d i n AutoD;

therefore

e

it

xee

it

(AutoD)x= (AutOD)x

f o r a l l t d R , so t h a t { e i t ; t d R } c W .

S i n c e W i s an a n a l y t i c

i s connected (1431 p r o p o s i t i o n 1 page 5 0 ) . Then t h e u n i t d i s c o f t i s c o n t a i n e d i n W , whence t h e r e s u l t s u b s e t of V , V \ W follows.

# 7.23. PROPOSITION. L e t D b e a b o u n d e d c i r c u l a r d o m a i n i n E.

T h e n , if E o n D is c o n n e c t e d ( t h u s , i n p a r t i c u l a r , if D is

b a z a n c e d l , we h a v e

CHAPTER

126

7

(AutD)O= (AutoD)O= R = E fl D 0

Proof: First we show that Rc(AutoD)O. Let xcR be given. By lemma 7 . 2 2 we have Oe(AutO)x; thus, gx= 0 for some gsAut0 D. -1 -1 Then we have x= g Ocg (AutoD)O= (AutO)O and Rc(AutoD)O. Now we show that (AutD)ncQ. Obviously O c R ; by lemma 7.21, R is invariant under AutD, so that applying AutD to the relation OeR we get (AutD)0 c (AutD)R c R.

Thus , we have R c (AutoD)0 C (AutD)O c R and corollary 7 . 1 8 completes the p r o o f .

#

COROLLARY. L e t D b e a bounded c i r c u l a r dom ai n i n E. T h e n , i f E 0 fl D i s c o n n e c t e d ( i n p a r t i c u l a r , i f D i s balanced), t h e orbit (AutD)O is b a l a n c e d . 7.24.

Proof: Let xe(AutD)O and ACE, I A l S l , he given. B y proposition 7 . 2 3 we have xeR; then by lemma 7 . 2 2 we have hxc (AutoD)X C(AutoD)(AutD)O = (AutD)0 .

#

.

0

54 .- The decomposition AutD= (Auk D) (AutoD) I -

7.25. THEOREM. Let D c E and 8=$ be bounded c i r c u l a r d o ma in s i n t h e Banach s p a c e s E and 2 r e s p e c t i v e l y , and assum e % % t h a t f: D+ D is an a n a l y t i c i s o m o r p h i s m of D o n t o D s u c h t h a t f(O)= 0 . T h e n , t h e r e is a s u r j e c t i v e c o n t i n u o u s l i n e a r map FcL(E,2) s u c h t h a t FID= f.

Gt:

' L ' L

Proof: For tdR we define gt: E+E and E+ E by means t 'L t %t of g (x)=: eitx and g ( y ) = : city. Obvioysly g eAutD, GtcAutD 'L and, as f : D+D is a surjective isomorphism, the mapping h=: g -t f-1'Lt g f satisfies heAutD and h(O)= 0. From the chain rule 'Lt % and the fact that we have Lgt= g L for all LeL(E,E) , we derive -t ( 1

hA1= ( g

lo

(f

-1

( 1 'Lt ( 1 -

lo

(g

lo - g

-t

-1

(f

(1 (1 t

lo

f

0

= id

so that, by Cartan's uniqueness theorem, we get h= idD and

127

BOUNDED CIRCULAR DOMAINS

e i t f ( x ) = f ( ei t x ) for a l l

ttB

and xcD. By developi,ng

both

Gtf and

fgt i n t o t h e i r

Taylor series a t 0 we o b t a i n

s i n c e w e have assumed t h a t f ( O ; ) = 0 . Thus, by t h e u n i q u e n e s s of t h e T a y l o r s e r i e s , l e ( n - l ) i t - l ] f o( n ( x , . . , x ) = 0

f o r t d R , 1112, and xeD; t h e r e f o r e f i n = 0 . f ( x )= f ive .

(1

0

(XI

f o r xcD and f = F

It f o l l o w s t h a t (1

ID

'L

where F=: f o c L ( E , E ) i s s u r j e c t -

# 7 . 2 6 . COROLLARY. L e t D a n d

E and

8

8

b e t h e o p e n u n i t b a l l s of

'L

and assume t h a t f : D+D i s a h o l o m o r p h i c map of I) o n t o

'L

D s u c h t h a t f ( O ) = 0 . Then f i s an i s o m o r p h i s m if and o n l y i f we

h a ve f = F

f o r some s u r j e c t i v e l i n e a r i s o m e t r y F : E+8. ID

P r o o f : Assume t h a t f i s a n isomorphism. By theorem 7 . 2 5

w e have f = F

'L

f o r some s u r j e c t i v e F e L ( E , 2 ) . S i n c e f ( D ) = F(D)=D, ID

F i s an i s o m e t r y . The c o n v e r s e i s c l e a r .

# 7 ; 2 7 . DEFINITION. L e t D be a bounded c i r c u l a r domain i n E. We d e f i n e t h e " i s o t r o p y s u b g r o u p " o f t h e o r i g i n , I s o t D, by means of

Obviously I s o t D is a c l o s e d subgroup of A u t D for t h e t o p o l o g i e s T and T a . Moreover, from c o r o l l a r y 7 . 2 6 w e immediately o b t a i n 0

t h a t I s o t D = Aut D. The c i r c u l a r subgroup Z s a t i s f i e s

Z C (Aut'D)

n

(AutoD).

CHAPTER

128

7

7.28. THEOREM. L e t D b e a b o u n d e d circuZar9 domain in E . T h e n , if E IlD is c o n n e c t e d ( i n p a r t i c u l a r , if D i s b a Z a n c e d i , 0 ue h a v e

AutD= (AutOD) (AutoD)= (AutoD)(Aut0D) Proof: Let feAutD be given. From proposition 7.23 we

derive f(O)e(AutD)O= ( A u t o D ) O ; thus we have f(O)= g ( 0 ) for some gsAutoD. Then h=: g-lfcAutD and h(O)= g-'f(O)= 0, whence by theorem 7.25 we obtain h= F for some surjective F a L ( E , E ) , so 0 ID that heAut D. Thus we have f = gh with gaAutOD and hsAut0D. The other equality comes from the fact that AutOD is a normal subgroup of AutD. Remark that the factorization f= g.h, with g and h in the above conditions, is not unique as the circular subgroup satisfies Z c (AutoD)I7 (AutOD)

.

§5.-

ff

Holornorphic and isometric linear equivalence of~Banach___ spaces

.

'L

Let E and 3 be complex Banach spaces and D, D their respective open unit balls. 7.29. DEFINITION. (a) We s a y t h a t E a n d 2 a r e " h o Z o m o r p h i c a Z Z y e q u i v a Z e n t " if t h e r e is some s u r j e c t i v e a n a l y t r ?i ic i s o m o r p h i s m f: D+D. ?i

(b) We s a y t h a t E a n d E a r e " i s o m e t r i c a Z Z y Z,LnearZy e q u i v a l e n t " i f t h e r e e x i s t s some s u r j e c t i v e l i n e a r i s o m e t r y L:

E+$.

7.30. THEOREM. E a n d 2 a r e isomorphically e q u i v a l e n t if and onZy i f t h e y i s o m e t r i c a Z Z y Z i n e a r l y e q u i v a l e n t . Proof: The "if part" is obvious. Thus, let f: "DD be any surjective holomorphic isomorphism. Then, the mapping PI, f#: AutD + AutD given by g-tf-lgf is a surjective isomorphism o f these groups, so that we have Aut8= f-l(AutD)€ and, therefore,

BOUNDED CIRCULAR DOMAINS

,

f (Aut8)= (AutD)f

129

(AutD)= (Aut8)f

f

-'

Let us denote by Eo and 80 the Banach spaces associated with E and 2 by definition 7.5 , so that

8

(AutD)O= E0 n %

,

(AutD)O= E o n D

(7.9)

by proposition 7.23. We claim that (7.10)

2.

(AutD)O = f-' (AutD)0

(AutD)O = f (Aut8)0, 'L

Indeed, from f-'(AutD)f= AutD we get that [f-1 (AutD)f]O=

'L

%

(AutD)O = E o

'

L

nD

'L

is a complex-analytic closed set in D. Let us put < = : f ( 0 ) ; % then f-l(AutD)< is a complex analytic closed set in D and, applying f we obtain that (AutD)< is a complex analytic closed subset of D. Therefore, by proposition 7.23 we have < c E0 n D. Then, as the orbit E o n D of 0 is AutD-invariant,

E" 0 n 8= (nut&)O = =

[f-' (AutD)f] O =

f-l (AutD)CCf-'(AutD) ( E o

n D) c f - l

( E o n D)

so that, applying f we obtain

In a similar manner we get E~ I

t

D c f (80n

8)

.

%

Thus we have E o n D= f ( E o n D) , which is equivalent to (7.10) 'L From the second of these formulas we obtain f - (0)E: (AutD)0 ; 'L % 'L 'L thus we have f-'(O)= q ( 0 ) for some gcAutD. Then h=: fg is an 'L

analytic isomorphism of D onto D with h(0) = fg(0)= 0 the result follows by corollary 7.26.

whence i+

CHAPTER

130

7

The group ~. __ of surjective linear isometries of a Banach space.

56.-

-

Let E be a complex Banach space with unit ball D=: B ( E ) . The group Aut 0 D of all surjective linear isometries of E turns out to be a subgroup of both AutD and GL(L(E)) I the linear group 0 of E. We obtain some properties of Aut D by looking at it a s a subgroup of these two groups. h

Let autD= autUDOaut0 D be the decomposition of the Lie algebra autD given by proposition 7 . 6 . For aeGL(L (El , let a# be the adjoint of CL (cf. definition 4 . 2 6 ) . 7.31.

PROPOSITION. Assume t h a t D Cs h o m o g e n e o u s . Then 0

we have t h e foZZouing c h a r a c t e r i z a t i o n of Aut D a s a s u b g r o u p

of GL(L (El ) :

Proof: Let asGL(L(E)) be such that acAutOD. It is an immediate consequence of the proof of proposition 7 . 9 (d) that aut D is a#-invariant(even if D is not homogeneous). 0

In order to prove the converse statement, we show first that the relation a # (autoD)c autOD

implies a# (AutoD)c AutOD

Indeed, let feAutOD be given; by Lemma 7 . 1 2 we can find Ak eautoD, k= lr2r...,nr such that

Therefore, by proposition 5 . 1 3 we have a f= afcl-'= It

[a(expAl)a-']0.

.O

[a(expAn)a-']=

= a (expA1)a ..oa (expAn)= exp(a#Al),..,exp # #

(a A )

a n

BOUNDED CIRCULAR DOMAINS

131

From the assumptuion a#(autoD)cautOD we get a#AksautoD for k = 1,2,..,n so that, again by lemma 7.12, exp(aA )eAutoD and, # k finally a#f= exp(a#A1)

0 .

.oexp(a#Ao)eAut0D.

NOW, let acGL(L(E)) be such that a# (autoD)C aut D. Since D is 0 assumed to be homogeneous, by proposition 7.23 it follows that

so that cx (D)= [a(Aut D)1 0 c (AutoD)O = D 0

A similar argument with a-'cGL(L(E)) gives a(D)= D. Thus, a is a surjective linear isometry, i.e., mcAut 0 D. 79 7.32'. COROLLARY. If t h e u n i t baZZ D o f E i s h o m o g e n e o u s , 0 t h e n Aut D i s a r e a l a l g e b r a i c s u b g r o u p of d e g r e e 2 of GL(L(E)) In p a r t i c u l a r , Aut 0 D i s a B a n a c h - L i e g r o u p f o r t h e t o p o l o g y of u n i f o r m c o n v e r g e n c e on D.

.

Let AcautOD be given. By proposition 7.6 A(x) = c-Q, (x,x)I

xeD

for some csE. Then a#A has the expression (cf. definition 4.26) (a#A)x= a(c)-aQ ( C X - ~ X a-lx) , , xcD Thus, again by proposition 7.31, a#A belongs to autoD only if. mQ (a-'xp-'x) = Qa(cl(x,x)

VxcD

if, and

132

CHAPTER

7

which i s e q u i v a l e n t t o

NOW,

f o r f i x e d CEE and xeD, t h e mappings L ( E ) X L ( E ) + E g i v e n

re s p e c t i v e l y by

a r e o b v i o u s l y c o n t i n u o u s homogeneous polynomials of d e g r e e 2 and 1 , and ( * ) can be r e f o r m u l a t e d as

w i t h C C E , X E D and acGL(L(E)). T h i s a s e t of e q u a t i o n s d e f i n i n g 0

A u t D a s a r e a l a l g e b r a i c subgroup of d e g r e e 2 of GL(L(E)).

Then, theorem 6 . 4 0 c o m p l e t e s t h e p r o o f .

# 7.33. homogeneous.

EXERCISE. Assume t h a t t h e u n i t b a l l D of E i s 0

Show t h a t , on t h e group Aut D , t h e a n a l y t i c

topology Ta, t h e t o p o l o g y of l o c a l uniform convergence 9' and t h e topology TU of uniform convergence o v e r D c o i n c i d e . 57.-

Boundary behaviour and e x t e n s i o n theorems.

W e r e c a l l t h a t i f D i s a bounded c i r c u l a r domain, t h e n any

AeautD

a d m i t s a unique r e p r e s e n t a t i o n of t h e form

where ceE

0

, LeL(E)

and Q c : E

+

E i s a c o n t i n u o u s symmetric

b i l i n e a r mapping. I n p a r t i c u l a r , A i s and e n t i r e mapping. 7.34.

DEFINITION.

Let AsautD b e g i u e n ; t h e n ue seL

T h u s t h e numbers p and C d e p e n d o n t h e d o m a i n D a n d on t h e

v e c t o r f i e l d AcautD. For any f i x e d EcE and AcautD, w e c a n c o n s i d e r t h e i n i t i a l v a l u e

BOUNDED CIRCULAR DOMAINS

133

Problem d = A[y(t)], 2

(7.11)

y(O)=

5

+5

whose maximal s o l u t i o n $ ( t ) i s d e f i n e d i n a domain dom

5

IR. O f c o u r s e , i f 5eD t h e n w e have

+

5

of

- E. 5-

( t )= ( e x p t A j 5 and dom+

7.35. LEMMA. L e t AsautD a n d SeE b e g i v e n . T h e n : (a1 dom+5=(-c-110g[l + d i s t ( S, O ) -l]

,

-'I)

C-llog [I + d i s t ( 5 , ~ )

( b ) F o r a n y xeD and a n y tdR w i t h I t l < G - l l o g ( l + l l C-x/l - I ) we h a v e

/I

+5(t)-+x 1 1 (Y and o n t y if G"XG2G"YG.

8.19. THEOREM. F o r e a c h AcB, t h e m a p p i n g

is a b i h o l o m o r p h i c autornorphisrn of B a n d we h a v e MA'=

M-A

Proof: Consider the power series of MA M I (xi= (x+A) ? (-A*x)"= A + A

n=O

m

c

n=O

u

(-I)~(I-AA*)xA*xA* .A*X ( 2 n + l l terms

Similarly

.

if

158

CHAPTER

m

M;(x)=

A+

c

n=O

(-I,)"xA*xA*

8

...

L--I

A*X(I-A*A)

(2n+l) t e r m s

Therefore, MA (X) ( 1 -A*A) = ( 1 -AA*)M i (X)

X€D

that is,

MA (X)= (1-AA*) -'MA

(X) ( 1-A*A) '=

( 1 -M*)'Mi ( X ) ( 1 -A*A)

-'

it immediately follows that M A ( X ) * = MA*(X*) holds f o r all XeB. Now we can show that M A ( D ) C E , i . e .

Thusl by applying the previous lemma with G= : ( 1 -A*A)

(8.17) Since

-'

( 1 +A*X) I (8.16) is equivalent to

(x*+A*) ( I - A A * ) - ' ( X + A I S

(I+x*A) (I-AA*)-~ (I+A*x)

AUTOMORPHISMS OF CLASSICAL BANACH SPACES

X*A ( 1 -A*A)-

'-x* 1 -AA * (

-1

A= o=

'

( 1-A*A) - A*X-A* ( 7 -AA* ) -

159

'x

as it can be seen from the corresponding power series expansions, the difference between the right hand side and the left hand side in (8.17) equals l-X*X which is a positive operator whenever / I X 11 ( 1 . Thus (8.16) is established. Clearly, the Potapov-MBbius transformation M A is holomorphic. So it remains to prove that M A ~ M - A =id, and it suffices to see that MA[M-A(Y)]= Y for all Y in some neighbourhood of -A. Now, the relation M (X)= Y is equivalent to A

that is ,

'

Mi (Y)= Ml [ ( 1 -AA* ) 'Y

I/ Y' (1

whenever

L

(8.20)

LeautOB( E ) < = > L is a J*-derivation on E 0

Proof of (8.19): Suppose that LcAut B ( E proposition 7.9(d) we see that

Thus

(LA)(LA)* (LA)= L (AA*A) for all ACE

)

. From

.

Conversely, suppose that L is a J*-automorphism. Then L( E ) = E . Moreover, for XeL(H) with X= JlXl, we have XX*X= J I X I ; therefore

so that

holds for all A c E . Thus 11 L 11 G I . But L-l is also a J*-automorphism of E and, by the open mapping theorem 1 L-leL( E ) Thus / I L- 11 G I and L is a surjective linear isometry of E .

.

.

Proof of (8.20) Suppose that Leaut0B( E ) , and define 0 G =: exptL for tdR. Then GtcAut B( E ) and s o , for each fixed A C E , we have t

AUTOMORPHISMS OF CLASSICAL BANACH SPACES

-

=

( -

=

(LA) A*A+A (LA) *A+AA* ( L A )

dt

lo

G~A)A*A+A(

dt

lo

G ~ A ) * A + A*A(

dt

lo

169

t G A)=

Conversely, let L be a J*-derivation on E and set again t 0 G =: exptL for tdR. We must show that, for tdR, GtcAut B( E ) , or equivalently, that Gt is a J*-automorphism of E For fixed A c E , we have

.

d dt

) G - (~G ~ A , ( G ~ A *,

G ~ A =) - L G ~ ( G ~ A , ( G ~ A *, ) G ~ A +)

*,

+ G - ~( L G ~ A , ( G ~ A )

) + G - ~( G ~ A , ( G ~ A *,

) G ~ A +) G - ~( G ~ A , ( L G ~ A *,

G ~ A +)

LG t A ) =

= - G - t ~ ( G ~ A , ( G ~ A *, )

G ~ A +) G - ~ L( G ~ A , ( G ~ A *) , G ~ A =)

o

for all A c E and tdR. Thus G -(~ G ~ A, ) ( G ~ A *, ) G ~ A =) G O ( G O A , 0

whence GtcAut B( E )

(GOA)

*,

GOA) = (A,A* , A )

. iy

8.31. EXERCISE. Let E l and E 2 be J*-algebras on H and suppose that L c L ( E l l E 2 ) is a bijective mapping. Then L is isometric if and only if it is a J*-isomorphism.

§6.-

Minimal partial isometries in Cartan factors.

Recall that an operator JcL(H) is called a p a r - t i a i ! isometry if, for some subspace H of H, the restriction J is an isometry 0

and J

1

= 0.

IHO

It is a well known consequence of the existence

IHo

of the polar decomposition that

CHAPTER

170

(8.21)

8

A i s a partial isometry < = > AA*A= A,

AeL(H)

From (8.21) we can easily obtain the following 8.32. LEMMA. LeL

h e a J * - a L g e b r a o n H. T h e n a n y

E

linear a u l o r n o r ~ p h i s mo f B ( E ) p r e s e r o e s t h e s e t o f partial i s o t n c t r i e s of

F

.

8.33. DEFINITION. Let.

E

b e n J * - a Z y e b r a and asrjtlme i h a t

c b' a r e p a r t ' i u l isometrics. L e t U S s e t Hk =: ixcH; I/ JkxII = Ilx 11 f u r k = 1,2. We s a y t h a t J 1 is a p a r a t uf J2 if H l c H 2 a n d J = J hie w r i t e

J ,J

1

2

21H1

J1<Jz

if J 1 is p a r t of J

lIH1

.

2'

Clearly, the relation < is a partial ordering on the set of non-zero partial isometries of E . The minimal elements with respect to are called m i r i i m a Z partial i s o m e t r i e s of E

.

We recall that a net ( A , ), in L ( H ) is said to be convergent I JCJ to A with respect to the weak operator topology if we have llm = for all x,yeH. We write Tw for the weak 3 1 operator topology on L(H). 8.34. THEOREM. If t h e J * - a Z g e D r a E is c l o s e d i n L(H) with r e s p e c t to T t h e n E i s t h e T W - c l o s u r e of t h e linear W' h u l l . of p( E ) and mp( E

)=

{AcE ;

AA*A=A+O,

A E*A= QAI

Proof: Let Aet' be arbitrarily fixed and suppose that

where A+P(A) is the spectral measure of l A / , is the polar decomposition of A (cf. I13 1 )

.

Consider the sequence Y =: n Y,+~= A Y ~ A,

n= 0,1,.

.

Since

AUTOMORPHISMS O F CLASSICAL BANACH SPACES

171

h o l d s f o r a l l ndN. T h e r e f o r e , i f t h e sequence of odd polyno-

mials ( p n I n m

i s bounded on [0,

p o i n t w i s e on LO,

1 1 A /I ]

11 A I / ]

and p,

converges

t o some f u n c t i o n J i , w e have

I n p a r t i c u l a r , JP[a,P]cE

f o r e a c h a,BdR

w e l l known t h a t each o p e r a t o r JP [a,P J

.

.

+

I

a,Pm,

However, it i s is a partial

i s o m e t r y , and t h a t T -

~i Bore1 function}= SpanCP[a,B]; a,BclR+}

W

T h e r e f o r e A b e l o n g s t h e T w - c l o s u r e of S p a n ( E ) . ?1

Suppose now t h a t J c m p ( E ) and l e t XeE A=:

% * ' L

be g i v e n . D e f i n e A by

'L

J X J . C l e a r l y A C E and k e r J c k e r A , whence

range(A1 c ( k e r A ) l c ( k e r 5 ) l = {xeH;

[I ?XI/

=

I / x [I 3

Thus, i f t h e p o l a r decomposition of A i s a g a i n A= J / A ] , t h e n

%

T h e r e f o r e w e have J < y J

f o r some yea,

'L

Iyl= 1 , and, a s J i s

2.

minimal, J = y J . But now f o r any s p e c t r a l p r o j e c t i o n P[a,P]

of

\ A \ , t h e o p e r a t o r J P { ~ , R J i s a p a r t i a l isometry c o n t a i n e d i n 'L

J= y J .

I t follows t h a t JP({II All

I)=

J . Therefore

#

172

8

CHAPTER

E

8.35. P R O P O S I T I O N . L e t T -elaced %w

Jcmp( E

)

be a n y J * - a l g e b r a

w h i c h is

a n d a s s u m e t h a t f o r any Jep( E ) t h e r e is s o m e %

B = (Span mp( E

sucrlz t h a t J < J . T h e n

)TTW.

Proof: L e t F be the family of the finite sums 2.

...+J,, ndN, of mutually orthogonal minimal partial isome'L tries Jkc m p ( E ) , lSk$n, i.e.

%

J1+

for all pairs k , & with k f k . Then, the linear hull of F is T -dense in E because, given any Jcp(E 1 , the net W is not empty and it is weakly convergent F(J)=: { ? e F ; J"<J} to J .

8.36. COROLLARY. In p a r t i c u l a r , if e v e r y Jcp( E 1 'L

'L

c o n t u i n s some Jcmp(E) s u c h t h a t dim range J < m

E

=

(Span mp( E 1 )-W

then

.

8.37. P R O P O S I T I O N . A l l Cartan f a c t o r s of L ( H ) a r e

T -cZosed. W

Proof: Given any operators R 1 , R l c L ( H ) and a conjugation Q on H , the mappings L ( H ) + L ( H ) given by

LR1R2

: X - + R ~ X R ~,

T

-

Q'

X+QX*Q

are T W -continuous. On the other hand, if Fk is any Cartan factor of type k, 1skS3, we have

F1= {XcL(H);

X=

XI

for some projectors P 1 , P 2 F 3 = { X s L ( H ) ; T X= -XI

Q

As for Cartan factors F4 page 334.

,

the proof can be found in 1 1 9 1 ,

ff

AUTOMORPHISMS O F CLASSICAL BANACH SPACES

8.38.

DEFINITION.

Given a J*-algebra

E

*

t h a t a n o p e r a t o r A c E i s m i n i m a l if A E A = EA.

for t h e s e t of m i n i m a l e l e m e n t s o f

173

o n H, we s a y

We w r i t e m ( E )

E .

It i s i m m e d i a t e t h a t the set m ( E )

J*-isomorphisms

8.39.

E

L of

THEOREM.

.

i s preserved by a l l Moreover, if E i s T w - c l o s e d , t h e n

L e t P1,P2 and Q b e , r e s p e c t i v e Z y ,

o r t h o g o n a l p r o j e c t o r s w i t h H . = : range P 1

c o n j u g a t i o n on H .

F = { X e L ( H ) ; P2XP1= X I

F3 = { X S L (H)

; QX*Q=

t h e corresponding Cartan f a c t o r s . C a r t a n f a c t o r of t y p e I V . m ( F ) = {f@e*; ecH1,

A2= 0 )

XI

-XI

Furthermore

,

Let F 4 be a n y

Then

fcHZ),

m(F,)= m ( F q ) = {AcF4;

( j = 1 , 2 ) and a

p 2 = I X C : L ( H ) ; QX*Q=

1

1

j

Denote b y

m(F ) = {e@(Qe)*; 2

{f@(Qe)*-e@(Qf)*;

eeH}

erfc.HI

provided t h a t dimF4>l.

Proof: I n g e n e r a l , f o r

e , f c H w i t h efoff, the t w o

relations

C a s e k= 1. From t h e -~

above remarks it i s e a s y t o check

t h a t , f o r e e H l and f e H 2 , w e have f P e * e m ( F 1 ) .

Conversely,

let

A c m ( F 1 ) w i t h AfO be g i v e n . T h e n A= P Z A P l and w e c a n choose some

CHAPTER

174

8

xsH such that P2APlx#0. In particular, the vectors e=: p l x and f=: P,Ae satisfy AefQ and A*ffO so that L

(Ae 0 (A*f)*#O Moreover f@e*cF1, and from the minimality of A we obtain

whence A= (XAe)@(A*f)* for some XeE.

Case -k=

2. Let ecH be given. Then Q[e@(Qe)*]*Q= eP(Qe)* so that e@(Qe)*cFZ and, as remarked at the begining of the proof e@(Qe) * e m ( F 2 )

.

Conversely, let Aem(F2) with AfQ be given, and choose eeH such that AefO. Since,F 2 is *-invariant, re0 (Qe)*] * C P z EA3A[eP(Qe) *JA= (Ae)@(A*Qe) * Moreover, from AcF 2 we derive A*Q= QA, whence A= f@(Qf)* with f = : XAe for a suitable A d . Case k = 3 . For the sake of shortness, we introduce the notation [v,u]=:

v@(Qu)*-LIB(Qv)*

v ,ucH

NOW, l e t f,eeH be given; it is easy to see that [f,eJcF3. For arbitrary AcL(H) Q[f,e]*Q= - [ f , e J , hence [f ,e]A* [f ,el= M-N holds with M= f@(Qe)*+eP(Qf)* N = fO(Qf *+e@(Qe) *

AUTOMORPHISMS OF CLASSICAL BANACH SPACES

175

Since F3 is a J*-algebra, for AeF3 we have [f,e]A*[f,e]eF3. Moreover, from AeF3 we obtain A*Q= -QA; it follows that M= [f ,el eF3 Therefore (8.22)

N= M-[f,e]A*[f,e]cF2 [f,e]A*[f,e]=

n F 3 = 101. Thus, for AeF3

[f,e]cC[f,e]

On the other hand, the relation [f,e]= 0 holds if and only if f,e are linearly dependent. By putting A=: [f,e] in ( 8 . 2 2 ) we obtain

[fie][f,e] * [f,e] =

A

[fie]

with A = /I ell 2 11 flI 2 -l<elf>121whence and [f,e]sm(F3).

[f,e]A*[f,e]=

C[fle]

Conversely, let Aem(F3) with AfO be given. For e,feH we have CA3A[f,e]*A=

(AQe)@(A*f)*-(AQf)@(A*e)"

Now we can choose e l f such that the vectors AQe and AQf are linearly independent. (Indeed, if we had dim range A= 1 , then A would have the form A= u@v* for some u,veH. But u@v*eF3 holds if and only if u@v= 0). Hence QAaf 'Be I *-e '@f * = for some independent couple f',e'eH

[f ' ,e '3 and A= A[f',e'].

# Case k = 4 . Suppose that AeF4 and A XeF4, we have

2

=

0. Then, given any

AX*A= (AX*+X*A)A= [(A+X*) 2-X*2]AeCA i.e., Aem(F4). Conversely, let Aem(F4) be given; then AF*A= CA and A 2 = a1 for some a d : .Thus, for some Bee, 4

CHAPTER

176

8

2 * 2

@A= (A*AA*)*A= A A A

= a2A*

If ct 2 f o r we have A*= yA for some yea. Since we have assumed that dim F4>l, we may fix X c F q \ Q A if afO. But then we get the contradiction

* * * *

2

2

QA3A(A X A ) A= A XA =

CI

2

X

Proof: Exercise 8.41. THEOREM. With t h e p r e c e d e n t n o t a t i o n s , 0

( a ) E v e r y e l e m e n t of Aut B(Pk)' k = 1,2,3, i s a c o n t i n u o u s

operator

w i t h r e s p e c t t o t h e T w - t o p o l o g y o n Fk.

( b ) We h a v e F k = (Span m p ( F k ) ) - T w f o r k= 1,2,3.

Proof: (a) Suppose that ( A ) is a net such that j jeJ TW - 1 i m A I, = A in F k I and let LcAutoB(Fk) and f,ecH be 1

arbitrarily fixed. We have to show that < L ( A , ) f ,e>-+ 3

Case k= 1: Write f'=: P 2 f and e l = : P 1e. Then f'Re'*em(F1) and we can find vectors f " e H 2 e"cH1' such that

For any pair vcHZ

ucHIr and any operator XcFl we have

AUTOMORPHISMS OF CLASSICAL BANACH SPACES

177

(V@U*)X* (V@U*)= VPU* Applying this, first to f'Pe'* and A, and then to fl'@e''* and L(A) , we obtain ,Ae ' >f '@e I *

(8.24)

( f 'Pe' * ) A* ( f 'Qe ' * ) = < f I

(8.25)

(f"@e"*)(LA)*(f"@e"*)= f"@e"*

,

As L is a J*-automorphism, from (8.23) and (8.24) it follows = c (f"Pe"*)(LA)* (fa1@eIt*)

whence, by cornpairing with (8.25) = 0 a n d a t+O

n e i g h b o u r h o o d W I C V of t h e o r i g i n s u c h t h a t G t ( W l ) C V f o r a l l

t , Itlc6. L e t u s s e t W 2 = :

J(W1),

i s d e f i n e d on

so t h a t J G t J - l

W2 f o r I t \ < & NOW, . l e t xeW2 b e g i v e n ; f r o m G-tO=

0 we d e r i v e

( j # G t ) x = ( J G t J - l ) x = JG t ( e x p C x ) O = J G t (expCx)G- t o =

JIGi(expcx)]o=

By t h e o r e m 4 . 2 8

w e have G

t

c

# x

where L # C = [ L , c ] = Y Y

= (exp t L ) # C ~ =

z

t" n !L

~

C

~

n= 0

c

(1

f o r yeE,

i.e.

Lo

f o r ndN. T h u s , ( J # G ~ ) ~ J[expC =

whence

lo=

exp t ~ A l x

( J # L ) x = L o( 1X . The r e s u l t f o l l o w s by t h e i d e n t i t y

principle.

#

205

BOUNDED SYMMETRIC DOMAINS

9 . 1 7 . LENMA. We h a v e [ C x , C y ] =

0 r

[QxrQy]=

0

f o r a22

x,ysE. P r o o f : For a n y U , V C E , c o n s i d e r

N U , v

= : [A, ,Av]

- [Aiu‘Aiv]=

[ C U + Q U , CV+QV] - [ i C U - i Q v

By c o r o l l a r y 9 . 1 0 w e have N

utv

r

iCv-iQ

eL, and therefore

since [ Q ~ , Q( 1~= I (~ Q( 1Q ~ , - Q ~( 1Q,);’=

and

i f s = 0 , l and c c E . I t follows t h a t

J#Nix,y

= 2[cix,Cy];1=

2i[Cx,C

] ( l = iJ#Nx

Y O

IY

i.e. Nix,y

by theorem 4 . 2 5 .

= iNx

Thus

IY

C L n(iL)c(autD) n ( i a u t D ) = {Ol

V

3=

CHAPTER 9

206

9.. 18

.,,PROPOSITION.

( a ) (J Cc)x= c

#

WE have

f o r all ccE a n d xcU

(b) F o r a l l ccE, t h e U e c t o r fieZd J#Q, is a c o n t i n u o u s h o m o g e n e o u s polynomial o f s e c o n d d e g r e e . Proof:

( a ) L e t c c E and xcU b e g i v e n . Then w e have

Whence , by lemma 4 . 2 3 w e d e r i v e

so t h a t J # C c i s a c o n s t a n t v e c t o r f i e l d of v a l u e c .

(b) L e t ceE, xcE and ycU be g i v e n . By t h e p r e v i o u s s t e p w e have x = ( J # C x ) y ; t h e r e f o r e

i s a v e c t o r f i e l d t o which p r o p o s i t i o n 9 . 1 6 a p p l i e s . Thus, w e

have I

.

Now, f o r x i n a neighbourhood of t h e o r i g i n , t h e segment [O,x] E by means of l i e s i n U and w e can d e f i n e + : [0,1] ( J # Q c ) t x . I t i s e a s y t o check t h a t + ( O ) = 0 , so t h a t by $(t)= ( 9 . 8 ) w e have -f

207

BOUNDED SYMMETRIC DOMAINS

which i s a c o n t i n u o u s homogeneous polynomial of second d e g r e e i n x . The r e s u l t f o l l o w s by t h e i d e n t i t y p r i n c i p l e .

56.- ______ The p s e u d o r o t a t i o n s on autD. ~

_

I

_

_

9 . 1 9 . D E F I N I T I O N . For t d R , we i n t r o d u c e t h e m a p p i n g s t

$ :

E autD

+

E autD in t h e f o l l o w i n g m a n n e r :

( a ) L e t AsautD b e g i u e n . T h e n A a d m i t s a u n i q u e r e p r e s e n t a t i o n A= A i L

w i t h Ace& a n d L c L a n d we d e f i n e L

$L:

A= A +L

+

A it +L e c

( b ) Now Qt may b e e x t e n d e d t o EautD b y c o m p l e x l i n e a r i t y

b e c a u s e we h a v e E a u t D = ( a u t D ) @ i ( a u t D ) , t h e sum b e i n g d i r e c t . I n o r d e r t o show t h a t $t i s a L i e a l g e b r a automorphism of a u t D , we introduce an a u x i l i a r y transformation.

9.20. R ~ :E

-+

by

E

For tdIR, we d e f i n e t h e m a p p i n g

DEFINITION.

R ~ :x

+

e

it

x.

L e t AcautD b e g i v e n and d e n o t e by J : V

+

U t h e neighbourhood U

of 0 and t h e isomorphism J g i v e n by d e f i n i t i o n 9 . 1 1 . Thus, A is u n i q u e l y determined by i t s r e s t r i c t i o n t o U and J # $tA i s a holomorphic v e c t o r f i e l d on U = J ( V ) . B e s i d e s , by p r o p o s i t i o n 9.18,

J # A i s a n e n t i r e holomorphic v e c t o r f i e l d ( a c t u a l l y , J # A

i s a polynomial of d e g r e e n o t g r e a t e r t h a n 2 ) so t h a t R t # J # A i s a l s o an e n t i r e holomorphic v e c t o r f i e l d and it makes s e n s e t o t

compare ( R # J # A ) I u

with

J#($tA)

Iu

. We

g e t t h e following

result 9 . 2 1 . PROPOSITION.

f a ) We h a v e R;J#A=

t

J#+ A for a l l tm

208

CHAPTER 9

a n d AeautD. ( b ) F o r all

td[R,

+t i s a Lie a l g e b r a a u t o m o r p h i s m o f

autD. ( a ) S i n c e any AcautD may be w r i t t e n i n t h e form

Proof:

A= Cc+Qc+Lfor some ceE and L c L ,

e q u a l i t y Rt J =J

#

#

#

+

t

it s u f f i c e s t o check t h e

i n these particular vector f i e l d s .

L e t c c E be g i v e n . By p r o p o s i t i o n 9 . 1 8 ,

JC C C i s c o n s t a n t ; t h u s

w e have

~ ~ c ~~ )# x( e=~

(J

~ c , e) ixt = (J

# c ~ )e ~ itc =

# and

Q, i s a homogeneous polynomial of second d e g r e e , w e have

As

(J#$

t Q,)X=

J#Q

it x= J# ( e e c

-it

Q ~ ) X =e

-it(~#~c)x

and t

t

( R # J # Q c ) x = (R

)

(1

-t J#Qc ( R - t x ) = eitJ#Q, ( e - i t x ) = e R x

= e

Since f o r LsL

,

-it

it - 2 i t

e

J# Q, ( x ) =

(J#Qc)x

J L i s l i n e a r , w e have

#

( b ) Obviously, $t i s an isomorphism of autD a s a v e c t o r s p a c e . By s t e p (a), f o r A l t A 2 s a u t D , w e have

BOUNDED SYMMETRIC DOMAINS

209

whence t h e c o n c l u s i o n f o l l o w s by t h e i d e n t i t y p r i n c i p l e .

ff L e t u s d e n o t e by

TI

j

,

j = 0,1,

the canonical projections

a s s o c i a t e d w i t h t h e d e c o m p o s i t i o n a u t D = L@Q.

9.22.

1. I

LEMMA. The norm

d e f i n e d o n a u t D by

\ A / = : max{II ( n . A ) h k l \ ; I

i s i n v a r i a n t under alZ t r a n s f o r m a t i o n s 4

j , k = 0,13 t

,

tdR.

Moreover, i t

d e f i n e s t h e n a t u r a l t o p o l o g y on autD. P r o o f : L e t u s s u p p o s e t h a t w e h a v e An TI,A

I n

\An]

+

+

/I

n.A

jA,l

+

0 i n a u t D , so t h a t

0 . Conversely, i f

I1

1

=

C

k=O

0 then

11

-+

0 i n autD; t h e n

(n.A )

(kll

I n 0

(kl\

" ( 71 j An o

j,k= 0 , l ; t h u s by t h e o r e m 5 . 6 , w e h a v e I T , A

+

I n

0; thus

-+

-+

o

for

0 ( j = 0 , l ) and

A = n A + n A + O . n O n 1 n

Moreover,

from

4 tA=

@

t

+L w e g e t (Ac+L)= A . lt e c

TI^^ t A =

t

4 noA

El$

t A=

4

t

TIA (k-

(k

f o r a l l a e a u t D . From d e f i n i t i o n 9 . 4 w e o b t a i n ( A X c ) O - k ( A c ) O f o r Ad!:,

which completes t h e p r o o f .

ff 9.23.

DEFINITION. L e t B b e a n e i g h b o u r h o o d of t h e o r i g i n

i n autD s u c h t h a t

( a ) t h e mapping A s N

-+

expAsAutD is i n j e c t i v e

(b) N i s i n v a r i a n t u n d e r a 1 2 t r a n s f o r m a t i o n s 4 Then we s e t G = : expN a n d d e f i n e yt: expA

-+

expQtA

t

,

tdR

.

CHAPTER 9

210

for> t m and AsN.

Observe that by lemma 6.47 and lemma 9 . 2 2 such a neighbourhood exists,

Our next task will be to extend the mappings Y ' : G +. G to the identity component AutOD of AutD. By lemma 7.15, any GcAut D 0 admits a representation of the f o r m G= G1G2..Gn with G = expA and AkcN for k = 1,2,..,n so that we could set k

k

.. (YtGn)

Y t G= : ( Y t GI)

The trouble is that the representation of G we have used is not unique. 9.24.

PROPOSITION. L e t G1,G2,..,GneG b e s u c k t h a t

.

G1o G20 . o G n= idD a T h e n fort a l l tdR.

we

have

t t t ( Y GI). (Y G2). ( Y G ) = idD

.

Proof: Let us write

for tdR. We begin with the following observation: Given any XcautD, we have G#tX= X

(9.9)

for a l l tdR. Indeed, by assumption, there are A k e N , k = 1,2,..p such that G = expA Write X = : @ tY where Y=: $-tXcautD and t is k k' kept fixed; then, by proposition 5 . 1 3 and lemma 5.14, we have

= $

t

(GI#..G,#)Y=

From ( 9 . 9 )

$

t

t

t

(G1..Gn ) # Y = @ (idD ) # Y = $ Y = X . t

we c a n deduce G = idD' Indeed, set

21 1

BOUNDED SYMMETRIC DOMAINS

Fh =: Gt+h(Gt)-l for t, hdR. B y (9.9) we have

Fix=

(9.10)

X

for all XcautD and h m . Now we show that the mapping hdIR FhcAutD h= 0. Let A= A +LcautD be fixed; since ccE continuous real-linear mapping, we have -f

d +tA= dt

1 lim ( + t + h ~ - + t ~lim ) = (A h

h+O

is Tderivable at A cautD is a

-A

c

h+O

+

e

it)=

i(t+h)

Moreover, by lemma 6 . 4 5 , the mapping A-texpA, AcN, is real analytic with regard to the T topology on AutD. Thus, considering the composed mapping t + + t +~ exp we get the T derivability of t

+

t

A

I

YLH with H= expA. Applying

t

..

this to each of the Y Gk = expGtAk, k= 1,2,. , n , by lemma 1 . I 5 we get the Tweak derivability of Gt. Therefore, for some neighbourhood B of 0, we have 1

(Gtfh-Gt) converges in the norm

(1 .( I B

or, equivalently, 1 (Fh-idD) converges in the norm

I / .I/ G~ ( B )

whence it follows that

1 (Ft-id ) = A t T lim h D h+O

CHAPTER 9

212

t

for some A eautD, s o that h

-+

Fh is T derivable at h= 0.

Then, theorem 4.28 entails 1

T lim

h (F#x-x)=

h+O

[A~,x]

for all XeautD, so that by (9.10) we have [At,X]= 0 for all XeautD. Thus, by proposition 9.13 O= A

t

=

T lim h+O

1

tth

[Gt+h(Gt)-l-Gt(Gt)-l]= T l i m h (G h+O

t -1

-Gt) (G )

d t whence Gt= 0 for all t d R , that i s , G is constant and Gt= G o= i dD' 9.25. COROLLARY. L e t G1,G2..GncG a n d H1,H2..H,&G g i v e n and a s s u m e t h a t G,oG20..oG = Hl0H o . . ~ H T h 8 n (Y~G,). ( y t c n ) = ( Y tH ~ ) . ( Y ~ H ln~o )r a l , l 2 tm. m

.

.

.

be

Proof: We need o n l y to observe that, if H= expA w i t h AeN , then t

(YtH)-l= (exp$tA)-l=exp(-$A ) = exp[lp =

t

t

t

(-A)]=

-1

Y exp(-A) = $ (H

)

. #

Yt:

9.26. DEFINITION. F o r tCR, w e d e f i n e t h e mapping AutOD AutOD by means of -+

Yt: Gl..Gn w h e n e v e r GI

..

(YtGl) (YtGn)

+

,..,GneG.

We know that AutOD=

u

G";

therefore, in view of the previous

n m

proposition, the mappings Y Moreover, we have

t

are well-defined on AutOD.

BOUNDED SYMMETRIC DOMAINS

(YtG)(Y-tG)= G

and

Yt(GH)= (YtG)( Y t H )

213

,

for G, HcAutOD and tdR. 9.27. EXERCISES. Consider the mapping RxAut0D given by (t,G)

+

Y'G.

+

AutOD

Show that t

(a) For fixed G, the application (t,G) -t Y G is a one-parameter group t

Y G is real analytic when AutOD is endowed with the analytic topology Ta Is it T continuous?. (b) The joint application (t,G)

-f

.

57.- The pseudorotations on D. - ____ We recall that, by proposition 9.8, D is homogeneous under the action of AutoD, so that D = {G(O);

GcAutoD}

D i n the 9.28. DEFINITION. For tdR, w e d e f i n e Tt: D f o l l o w i n g m a n n e r : L e t xcD be g i v e n ; t h e n we h a v e x= G(0) f o r some GcAutoD, a n d we s e t -+

In order to see that this definition makes sense we have to verify that, for G1,G2eAut D with G1O= G20, we have (YtG )0= (YtG2)0. By passing to G=: G I 1G2 we must prove that, 1

t for all GeAutoD, the relation GO= 0 implies (Y G)O= 0 for all

tdR. This will be our next task. 9.29. DEFINITION. We s e t IsotD=: {GcAut D; So

GO= 0 )

far, we have made no u s e of the assumption concerning the

s i m p l e c o n n e c t i v i t y of D. We shall apply it to prove the

following:

CHAPTER 9

214

9.30. PROPOSITION. A s s u m e t h a t t h e b o u n d e d s y m m e t r i c d o m a i n D i s s i m p Z y c o n n e c t e d . T h e n t h e s u b g r o u p IsotD is a r c w i s e c o n n e c t e d v i t h r e g a r d t o t h e t o p o Z o g y Ta.

Proof: It suffices to show that, for any GsIsotD, there continuous path :'I [OJ] IsotD such that r ( O ) = idD and l'(l)= G.

exists a T

-f

Let GeIsotD be given; then we have GeAut D

and

GO= 0.

Therefore, we can find A1,A2,...,A,cN

We divide the interval

k for te [ 5

, k+l

1

such that

[O,l] into n subintervals

and k = 0,l , , , ,n-I . Obviously,

f

is a

T a continuous path which connects id and G in the space AutoD. D

In order to connect them in the subspace IsotD, we project this path f : [ O r I ] + AutOD into D by applying each ? (t)= G to the origin 0, so that we get the path y: [0,1]

t

+

D defined by

Since G belongs to IsotD, y is a closed path: y(O)= id,(O)= 0 and y ( l ) = G ( 0 ) . Thus, as D is assumed to be simply connected, y is homotopic to the origin 0. Let us denote by R=: [O ,I] x [ O r 11 the unit rectangle and denote by f: (s,t)eR + f(x,t)eD a homotopy in D continuously deforming the path y into the origin 0, so that we have

BOUNDED SYMMETRIC DOMAINS

215

f(O,t)= y(t)= Gt(0)

f(l,t)= id,(O)=

f ( s , O ) = idD (O)= 0

f(s,l)=

0

(9.11) G ( O ) = 0.

We shall construct a lifting of f: R + D to AutoD, i.e., a T continuous function f: R + AutOD such that

for all (s,t)cR. Then we shall have F(l,t)O= f(l,t)= 0 for all tc[0,1] , so that F(l ,t)eIsotD for te[0,1] and, by writing

we obtain the path

r:

[0,1]

+

IsotD we were looking for.

Let U be the neighbourhood of the origin in E constructed in the proof of proposition 9.8; thus the mapping (9.12)

g: ceU

-+

(exp Ao)Oeg(U)

is an isomorphism and

Then, we have

for some carathgodorian open ball B E ( 0 ) centered at 0. As the homotopy f: R -+ D is continuous, the mapping

is uniformly continuous on RxR; therefore, there exists an m a such that

CHAPTER 9

216

Now we devide the horizontal side [ 0 , 1 ] into m subintervals

of the rectangle R

and construct recurrently the lifting F of f on each of the k+l subrectangles R =: [ k , ]X[O,l], k = O,l,..,m-l.

m

k

We claim that, for (s,t e R 1

, we have

G;

Indeed, as the caratheodorian distance is AutD-invariant, by

(9.11) we have

Therefore, by ( 9 . 1 2 ) and we define

it makes sense to apply J-l to G;lf(s,t)

c(s,~)=:J - 1 Gt-1 f(s,t)

for (s,t)eRo. Let us set F0(srt)=: GtexpA

c(s,t)

for (s,t)eRO. Then, it is easy to check that Fo is a lifting of f over R o . Now we proceed by induction on k. Assume we had already

constructed a lifting Fk of f over R ; thus

for all (srt)CRk. We claim that, for ( ~ , t ) e R ~,+we ~ have

217

BOUNDED SYMMETRIC DOMAINS

Indeed, by ( 9 . 1 3 )

and the induction hypothesis we have

Thus, it makes sense to apply J-l to Fk- 1 ( ;k;i ,t)f(s,t) and we define

for (s,t)CRk+l.If we set Fk+l(slt)=:Fk(

mk

,t)expA c(s,t)

for ( s , t ) ~ R ~, +then ~ it is easy to check that F k + l lifts f on R k + l . Moreover, Fk+l and Fk agree on the common border of their rectangles of definition:

so that Fk+l extends the previous partial lifting. This

completes the proof.

# Let U and V be the neighbourhoods of 0 in E constructed in definition 9.8 and put R ~ = :x

-t

e itx

for tdIR and xeE. We may assume that U is an open ball centered t at 0 , so that U is invariant under the transformations R By setting

.

it is easy to see that V is invariant under the transformations St. Finally, we recall that, by proposition 9 . 2 1 , we have

21 8

CHAPTER 9

€or a l l AcautD. 9.31.

LEMMA. L e t t d R b e g i v e n .

T h e n , t h e r e a r e a number

6 > 0 and a n e i g h b o u r h o o d W of 0 s u c k t h a t we h a v e

P r o o f : Take a n y 6 > 0 s u c h t h a t t h e b a l l B 2 & ( 0 ) w i t h c e n t e r a t 0 and r a d i u s 2 6 i s c o n t a i n e d i n V;

then

S t [ B 6 ( 0 ) ] C V i s a neighbourhood o f 0 and w e d e f i n e

The p a i r 6,W s a t i s f i e s o u r r e q u i r e m e n t s .

11

I n d e e d : L e t AcautD b e s u c h t h a t

All

and t a k e any x e B 6 ( 0 ) .

Consider t h e i n i t i a l v a l u e problem d dt

y ( t ) = A[y(t)]

I

y(o)= x

whose s o l u t i o n i s d e n o t e d by y ( t ) = ( e x p t A ) x , a n d s e t T(x)=:

inf{t>O;

(1

(exptA)x-xl(

W e claim t h a t , r ( x ) > l . Indeed, f o r OSt