SPECTRAL THEORY AND COMPLEX ANALYSIS
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NORTH-HOLLAND MATHEMATICS STUDIES
4
Notas de Matematica (49) Editor: Leopoldo Nachbin
Universidade federal do Rio de Janeiro and University of Rochester
Spectral Theory and Complex Ana1ysis
J E A N PIERRE FERRIER University of Nancy I
1973
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
0 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM - 1973
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.
Library of Congress Catalog Card Number: 72 93089 ISBN North-Holland : Series: 0 7204 2700 2 Volume: 0 7204 2704 5 ISBN American Elsevier: 0 444 10429 1
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P R I N T E D I N THE N E T H E R L A N D S
INTRODUCTION
Th es e notes a r e issued from l e c tu r e s given by the author a t the "Collkge d e F r an ce" in 1971, the purpose of which was an exposition of complex analysis in Cn based on s p ect r al theory. Such a n approach leads to global theorems in connection with holomorphic convexity, approximation problems or ideals of holomorphic functions, and makes possible the introduction of growth conditions. It is eas y to apply the holomorphic functional calculus of Banach al g eb r as to polynomial approximation of holomorphic functions on a neighbourhood of a polynomially convex compact set K in C", by proving that K is the joint spectrum of the coordinates in the closed subalgebra generated by the polynomials i n e ( K ) , This method
-
leads to the so-called Oka Weil theorem. We r e m ar k that polynomial convexity is equivalent to the existence of a family (p, ) of polynomials such that
K(')
(1)
=
IP,(s)l
for e v e r y s in C", where fK denotes the c h a r a ct er i st i c function of K . A s t h e holomorphic functional calculus only r e q u ir e s convexity with r e s p e c t to the r e s t r i c tions of polynomials to a given neighbourhood of K , an improvement co n si st s in asking for condition (1) when s belongs to such a neighbourhood. It would be more difficult, however, to u s e the theory of Banach al g eb r as in proving the same r e s u lt when polynomials are replaced by the al g eb r a O(n)of holomorphic functions on a given pseudoconvex domain f l , and polynomial convexity by convexity with r e s p e c t to or the family (pa) by a family (fa) of su ch Moreover, the r e s u l t is t r u e upon replacing Ifa\ by a positive function X, on that log IT, is plurisubharmonic, as it h a s been proved by L. Harmander. All theorems mentioned above concern approximation on K of functions defined on a neighbourhood of K , or approximation for the compact open topology. It is possible to obtain better r e s u l t s by considering growth conditions, but the al g eb r as of holomorphic functions will no longer be Banach a l g e b r a s . W e sh al l t h er ef o r e u s e spectral theory of b al g eb r a s , as introduced by L. Waelbroeck. A sufficiently g en er al setting, including al l cl as s i c a l examples, is the following : if is a non negative function on Cn such that Is(s)- 8 ( s ' )I ,C 1s - s ' \ for all s , s ' in Cn and Isl8(s) is uniformly bounded, we consider the algebra z(8) of complex functions f on the open set )8zO) N such that 6 If 1 is uniformly bounded for some positive integer N, and the subalgebra 6(&) of all functions of qs)which are holomorphic. Elementary p r o p er t i es of su ch a lg eb r as are given i n Chapter I. W e s a y that two non negative functions 8,, 6, on Cn
o(fl),
-
c)(fl).
v1
INTRODUCTlON
<
"N N a r e equivalent if c d 1 < 8, and c S 2 8, for some positive integer N and some & > O ; if 8,, 6, a r e equivalent, the algebras o(8,) are equal. Let 8 , 8 be two non negative functions on Cn satisfying the required properties. If St2 & and i f 18' > 0 ) ~ i connected, s the algebra o(8') can be considered as a subalgebra of fJ(8). We s a y that s') is dense in i f , for every function f of there e x i s t s a sequence fn in 8 ) and a positive integer N such that S N \ f n - f 1 tends uniformly to zero. The basic approximation theorem, proved in Chapter VI, s t a t e s that i f {(? > 0) is assumed to be pseudoconvex and, up to equivalence, -logs plurisubharmonic on $>O\, a necessary and sufficient condition for density of 8 ) in o(8)is the existence of a family (f,) of 8') satisfying, up to
a(
o(x),
a(s,),
@(s)
o(
a(
a(
equivalence, the relation 1/s = SUP
(2)
ifb\
18 > 01 ( r e s p . Idlo]). Moreover, the r e s u l t is still valid upon replacing f, by a positive function IT, of 6 ' ) such that log is plurisubharmonic. We apply our theorem to approximate holomorphic functions on a given domain by polynomials o r holomorphic functions defined on a l a r g e r domain or satisfying more restrictive growth conditions. We discuss the Runge property through this method i n Chapter VI , The approximation theorem is not trivial when 8 = 8 When 8 is the distance to on
a
.
the boundary of an open s e t , an e a s y consquence is the fact that every pseudoconvex domain is a domain of holomorphy ; moreover, there exists a n holomorphic function with polynomial growth which cannot be extended to a l a r g e r domain. Actually, these properties a r e deduced in Chapter IV from a theorem of I. Cnop concerning the joint spectrum of the coordinates i n the algebra a(8). This s p e c t r a l theorem a p p e a r s as a particular case, but with parameter, of the corona problem for algebras of holomorphic functions with restricted growth, which has been solved by L. Hijrmander. Conversely, spectral theory has applications to the theory of ideals in algebras of holomorphic functions, which are given in Chapter V . Instead of we consider, which are more generally, algebras of holornorphic functions on a given domain inductive limits of algebras W e prove for such an algebra A the following decomposition property : i f f vanishes a t a point s of , we have
o(S),
a(&.
...
n
f = (Z1-S 1) g 1 +...+ ( z n - s n ) g n ,
where gl , ,gn can be chosen in a bounded set of A when f varies in a bounded set of A and s i n W e d i s c u s s holomorphic convexity for algebras satisfying the decomposition property and give a characterization of inductive limits of algebras 06). Our exposition heavily depends on the estimates of L. Hijrmander for the 3 Neurnann problem, but only through the spectral theorem. All other properties are deduced by means of spectral theory and functional calculus. In Chapter VII , however,
n.
vi j such estimates a r e used again to obtain new spectral and approximation theorems. We apply these last results to study plurisubharmonic functions on a pseudoconvex domain and give a generalization of a theorem of H. Bremermann which is independent of methods of Hartogs The study of algebras of bounded holomorphic functions on open domains is not suitable for spectral methods. In a quite different direction, it seems very difficult to introduce growth conditions in complex manifolds and develop L2 methods there. The case of relatively compact domains of complex manifolds have been studied by R . Narashiman, but i t is not very different from the C" case, because of the existence of a finite number of local maps, The original exposition started from spectral theory of b - algebras and the unsimplified holomorphic functional calculus of L. Waelbroeck. It is not likely to assume that t h e reader knows everything about such a construction, Therefore, as an introduction to the theory, all definitions and properties which a r e essential for the following chapters a r e given in Chapter 111, with a short construction of the functional calculus. The reader is referred to more complete texts for the multiplicative property. W e discuss, however, the easier case of Banach algebras. Algebras of holomorphic functions with restricted growth a r e not Banach algebras but directed unions of Banach spaces. It would have been possible to consider on such algebras the direct limit locally convex topology but bounded structures are more natural and lead to more precise theorems. Basic definitions about such structures a r e given in Chapter 11. A few elementary properties of plurisubharmonic functions and pseudoconvex domains a r e also recalled in Chapter IV. The basic estimates for the 7 Neumann problem are however admitted. The reader can find more information on these topics in the first chapters of Hormander's book.
.
-
The original lectures were given in French; a s I am not well-acquainted with the English language, the translation I have written on Publisher's request is probably awkward. However, I hope these notes w i l l be useful to those who are looking for an introduction to a new aspect of complex analysis which is suitable for a large development. Jean - Pierre Ferrier College de France, Paris January 1971 and University of Nancy October 1972
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CONTENTS
INTRODUCTION L I S T OF SYMBOLS
........................................... ........................................
V
xi
CHAPTER I . .ALGEBRAS O F HOLOMORPHIC FUNCTIONS WITH RESTRICTED GROWTH
....................................... Weight functions ........................................ Elementary p r o p e r t i e s ..................................
1 . 1 . B a s i c definitions 1.2. 1.3.
........................ 1 . 5 . Inductive limits ......................................... Notes .................................................. 1 . 4 . Regularization of weight functions
1 2
3
5 8 11
CHAPTERI1.- BOUNDEDNESS ANDPOLYNORMEDVECTOR S P A C E S
............................... ............................... 2 . 3 . Completeness ........................................... 2 . 4 . C l o s u r e and density ..................................... 2 . 5 . Algebras and ideals ..................................... Notes ..................................................
2 . 1 . Polynormed vector s p a c e s
12
2 . 2 . Convex bounded s t r u c t u r e s
14
CHAPTER I11
16 18 18 21
.- S P E C T R A L THEORY O F b-ALGEBRAS
3 . 1 . Spectrum of elements in a Banach a l g e b r a ................. 3.2. S p e c t r a l sets ...........................................
23
3 . 3 . S p e c t r a l functions
25 28
3.4. The holomorphic functional calculus
30
3.5.
37 39
...................................... ....................... .......................... S p e c t r a l theory modulo a b-ideal Notes ..................................................
CHAPTER1V.- SPECTRALTHEOREMSANDHOLOMORPHIC CONVEXITY
........................................... O(S) .............................. 4 . 3 . S p e c t r a l functions for z in o(&) .......................... 4.4. Plurisubharmonic regularization ........................... 4.5. Domains of holomorphy .................................... 4 . 6 . Bounded multiplicative l i n e a r forms ........................ Notes ................................................... 4.1. Preliminaries 4 . 2 . S p e c t r a l s e t s for z in
41 43 47 49
51 52 53
CONTENTS
X
CHAPTER V ..DECOMPOSITION PROPERTY FOR A I L E B R A S OF HOL.OMORPHIC FUNCTIONS
5 . 1 . l’reliminaries .......................................... 5 . 2 . Decomposition property for (3(g) ........................ 5 . 3 . Decomposition p r o p e r t y for s u b a l g c b r a s ................... 5 . 4 . S p e c t r a l functions for z ................................. 5.5. Convexity with r e s p e c t t o a l g e b r a s of holomorphic functions .. 5 . 6 . I d e a l s of holomorphic functions Notes
...........................
.................................................
CHAPTER VI .- AF’PROXIMATION THEOREMS ............................ 6 . 1 . Approximation on compact s e t s 6.2. Hunge domains and g e n e r a l i z a t i o n s
.......................
6 . 3 . B a s i c approximation theorem ............................. 6.4. Approximation with growth ............................... Notes ..................................................
CHAPTER VI1.- FlLTRATIONS 7.1 Filtrated b algebras
.
-
.................................... ...........................
7 . 2 . S p e c t r a l theorem with filtration
7.3. Application t o plurisubharmonic functions 7 . 4 . Approximation theorems with filtration
59 61 63 64 67 68
70 72 73 76 77 78
..................82
.....................
............................ Notes ................................................. BlBLIOGRAPHY .............................................
7 . 5 . Polynomially convex open s e t s
55 58
84 88 89 91
LIST O F SYMBOLS
is the c h a r a c t e r i s t i c function of A . JA k e r f is the kernel of the l i n e a r mapping f [A is the complement of A. A , A, aA are respectively the c l o s u r e , the i n t e r i o r , the boundary of A.
.
-
z is t h e identity mapping of C",
z . the coordinate of index j . J
> 01 , where f is a real function on C" denotes f(s)> 0 . i z I is the Euclidian norm in C".
{f
the s e t of all s E C n s u c h that
d ( s , A) is the Euclidian distance f r o m a point s t o a s e t A. dX denotes the Lebesgue measure d" is the differential form a/az, d z l +. .+ a/dZn d z n .
.
(3(n)is the a l g e b r a of holomorphic functions in
c$),6(&) are defined in Section 1 . I . ( A ) , (3(A) are defined in Section 1 . 5 . z(8; E ) , Gr(8 ; E), Gcr( 0 ; E ) , o(s; E )
.GCB), .Gr($),
so
are defined in Section 2.1.
defined in Section 2.1.
2 -1/2
(l+lzl ) is defined in Section 1 . 2 .
=
Sfl
06)are Nc(f:r($),
0.
6 ,?
9
A,
~ ( s =) inf
S€C"
9
A,,
Aq,
+
(F(sl)
a r e defined in Section 1.5.
(see Section 1.4).
lsl-sl)
E D is the vector s p a c e spanned by B equipped with the Minkowski functional of 8 . F, when F is a v e c t o r s u b s p a c e of a b - s p a c e E , is defined in Section 2 . 4 . $ is defined in S e c t i o n 2 . 3 . AIXI, id1 (a,,
.. . , Xn ] is defined in S e c t i o n 2 . 5 , .. . , a n ; A) is the b - ideal generated by
a?,
.., , a n
in the b - a l g e b r a A (see
Section 2.5).
..,an) = s p ( a , , . . . , a,;A) is the spectrum of .. . , a n ) in A ( s e e Section 3 . 1 ) .
s p ( a ) = s p ( a ; A) ( r e s p . s p ( a l , . t h e joint spectrum of a ? ,
a (resp.
xii
LIST OF SYMBOLS
.. .
u(a) = u (a; A) ( r e s p , u ( a , , . .. ,a n ) = 6(a,, , arl; A)) IS t h e s e t of all s p e c t r a l s e t s for a ( r e s p . a , , . . . , a n ) in A ( s e e S e c t i o n 3.2). A ( a , , . . . , a n ) = A(a,, . . . , a n ; A) is t h e s e t of all s p e c t r a l functions for a l , . , a n III
..
A (see S e c t i o n 3.3).
A ( a ; A/I) is defined i n S e c t i o n 3.5. is defined i n Section 3.4 ( t h i s is t h e holomorphic functional c a l c u l u s a t a).
f [a]
d~
dz1 A
=
< x, y )
n
...
h
dZr,, d" u
replaces x , y ,
=
+. .
d"u
1 .+ xny,
A
., ,
A
d"url.
is t h e hull of K with r e s p e c t t o p l u r i s u b h a r m o n i c functions i n
KfZ
4.1).
is defined i n S e c t i o n 4.4.
"3 is defined i n S e c t i o n 5 . 1 . 8, is defined i n S e c t i o n 5 . 3 .
sB is defined i n S e c t i o n 5 . 1 . EH, K p , K a a r e defined i n S e c t i o n 6 . 1 . A
Kp
A
A
is defined i n S e c t i o n 7 . 3 .
v ( B ) i x t h c f i l t r a t i o n of B ( s e e S e c t i o n 7 . 1 ) . Xr, are defined i n S e c t i o n 7 . 5 .
A,,
(see Section
CHAPTER I
ALGEBRAS O F HOLOMORPHIC FUNCTIONS WITH RESTRICTED GROWTH
z((8)
W e define the algebra of tempered functions with respect to a weight function 8 , and the subalgebra of holomorphic functions of Weight functions a r e non negative functions on C" such that IslS(s) is uniformly bounded on Cn, satisfying 18(s)- 8(sl)l Is-sll for all s, s ' in C". Examples of such algebras are given, including the algebra of polynomials, entire functions of exponential type, entire functions of finite o r d e r or holomorphic functions with polynomial growth on an open set. A m o r e general condition on weight functions is introduced, which actually leads to the same algebras; moreover, it can be assumed that each weight function 8 is on the set @>O) W e study inductive limits of algebras (36)and the algebra of all holomorphic functions on a domain 0
as).
a(&)
<
.
c"
.
. .- Basic definitions
1 1
Let 8 be a non negative function on C"; we associate to 6 the s e t (s>O)where does not vanish and define 6 - tempered functions a s complex valued functions on {8> 0) such that 6N 1 f 1 is uniformly bounded on {8>O) for some positive integer N Thus 8- tempered functions a r e functions on $ > O ) which a r e bounded by a positive multiple of some negative power of 6 The s e t of all 8- tempered functionsis an algebra, which will be denoted by G((F). W e shall say that two non negative functions Sl, 8, on C" a r e equivalent if there exist a positive integer N and E>O such that
8
.
.
ES;<~,
and
ESy<S1
.
Equivalent non negative functions give obviously rise to the same set and the same algebra of tempered functions. More generally, if Es;y_( 8, for some positive integer
HOLOMORPHIC FUNCTIONS WITH GROWTH
2
> 0) contains the set
N and some &>O, the set
isl > 0) , and the restriction map-
ping is an homomorphism from a s 2 ) to n81). We suppose now that 6 is a non negative function on Cn such that \8>0] is open. For every positive integer r , we define as the algebra of all complex-valued functions on {6>0] such that every derivative of order s < r of f is 8-tempered. We also define o(8)a s the subalgebra of which consists of all functions in c(8)which are holomorphic on !8>Of In other words
rr($)
.
c((s)
where (3(a) denotes the algebra of all holomorphic functions in the open s e t 0 ; we recall that holomorphic functions on 0 a r e exactly locally integrable functions on satisfying d'Y = 0 in the sense of distributions, where d" is the differential operator
a/aq +...+ a/aq, . 1 . 2 . - Weight functions
In order to prove nice properties for 0(8), w e introduce restrictive conditions Precisely, if z = (zl, ,zn) denotes the identity mapping of Cn and Is1 = (Is,[ 2 +...+ lsnl2 )21 the hermitian norm of s
on
6
.
...
Definition 1
.-A non negative function 6 op Cn is called a weight function i f 8 e-
fies the following conditions : 111 1 )
1218 is uniformly bounded on
~ 2 16(s) ) - S(sl)l< 1s
- S'I
Cn.
for all s ,
sl
c".
Condition W 2 implies that 8 is continuous. For a continuous 8 , condition W 1 means that 6 = O( 1/12 I ) a t infinity. Hence contains 1, z l , . , , zn and therefore all polynomials Condition W 2 is deeply connected with spectral theory, a s we shall see in Chapter 111. On condition W 2 also depend all the properties proved in the following section.
as)
-
.
W e consider now a few examples of weight functions which lead to classical algebras of holomorphic functions 1) Let
so = c(
( 1 +1z12+.
8,) consists of all complex valued functions defined on Cn which have The algebra polynomial growth at infinity. Thus 8,) is the algebra of polynomials, because of
a(
ELEMENTARY PROPERTIES the theorem of Liouville. W e note that, for a continuous positive multiple of 6,. 2) If fying
8 = e-"',
the algebra
8 , condition
3
W 1 means that
$ is bounded by a
as)consists of all entire functions f on C" satis-
e-Nlz'f = o ( I ) , for some positive integer N, that is f =
Thus
O(e-"')
o(eNlzl).
is the algebra of entire functions of exponential type.
3 ) More generally, i f k is a positive integer and E >O so that &e-lzlk satisfies ee-Izlk) consists of all entire functions f satisfying f = o ( eN Izlk)
W 2 , the algebra
o(
f o r some positive integer N, that is the algebra of entire functions of finite order k . 4) Let 0 be an open set in C". W e associate to ned on Cn by
L(s) = Min ( &(s), d(s, 10 1,
[a
n
the weight function
&o
where d(s, ) denotes the distance from s to the complement of in @( ) are called holomorphic functions with polynomial growth on
1.3.
6n
defi-
0 , Functions
a.
- Elementary properties
We consider in this section a non negative function 8 on Cn satisfying condition W2 (for instance a weight function), For s, s ' in C" such that 1s - sll,< + 6 ( s ) ,we have I$(sl)-6(s)I +s(s) and
<
Such an easy property will be often used,
.-
Proposition 1 k t 8 be a non negative function on C" satisfying condition then each a/dzj m n s O($)iLoO(&). In other words, we have
m\
w 2;
o(6)c rr(8) for every positive integer r.
as);
Proof. Let f be a function of there exists a positive integer N such that is uniformly bounded. W e have to prove that bf / b z . satisfies a similar estimaJ te. Let s be a point in {8>0); we consider the polydisc D with center at s and radius r = 8 ( s ) / 2 6 . Obviously D is contained in the ball B with center a t s and
HOLOMORPHIC F U N C T I O N S WITH GROWTH
4
<
radius ) $ ( s ) . Hence, f o r i n D , we have j 8 > 01 Using Cauchy s formula, we get
.
8(<) >, f s ( s )and
D is contained in
and
Hence
af
-(s)
aZ.J
=
( 2 n ) "r
If M is a uniform bound f o r
for
{2?..
itl
raf(s+r(e
itn
, ... , e
-it. )) e J d t l
... dtn .
0
6N I f 1 , a s
< in D , we get
that is
and the result is proved, as N and M a r e independent of s , Let dX denote the Lebesgue measure in C", and p be a positive number. Proposition 2.- Let 8 bea weight function on Cn; the algebra holomorphic functions f 0" > O} satisfying
{ !f(s)IP sN(s)dX(s.1 c
O(s) is the se t of all
+ co ,
for some positive integer N. Proof. W e suppose f i r s t that f belongs to assuming < 1,
s
Sm\,l, we g e t ,
/(f(s)IP
for N 1 > pN
SN1(s)d)c(s)
+ 2n + 2 .
As
< Mp
82n+2= O(
a(6).If
M is a uniform bound for some
l J 2 n + 2 ( sdh(s), ) the right side of the inequality is
REGULARIZATION O F WEIGHT FUNCTIONS finite. Conversely, if
I
If(s)(
S N b ) d&)
<
+
00
5
,
f o r some positive integer N , and i f f is holomorphic, we consider for s in \ 8 7 0 1, the ball B with center a t s and radius 5 8 ( s ) . A s >, 5 J(s) for in B, the ball B is contained in 18 >O] and from subharmonicity of If Ip, we get
<
s(c)
and
N Upon replacing $ (s) by result follows f r o m
8N(c)i n the integrand, we only lose a multiple 2N and the
1 /vol(B) = 0(1/82n(s)).
1 ,4.- Regularization of weight functions
Any non negative function define first
6
8 on
C" gives naturally rise to a weight function. We
by
-
8
Obviously 8 satisfies condition W 2 and is smaller than 8 . Precisely is the largest function with such properties. In o r d e r to get a weight function, we only have to I
-
consider Min (&,8). A s algebras associated to equivalent functions are the same, all properties proved
in Section 1.3 a r e valid when 8 is only equivalent to a weight function. W e give now a necessary and sufficient condition for this Proposition 3.- Let
I$
be a non negative function on C"; in o r d e r that
6
is equiva-
lent to a weight function, it is necessary and sufficient that the following conditions a r e fulfilled : H 1) Restrictions of polynomials belong to N H 2) There exist a positive integer N and E 7 0 such that Is s1I<E8 ( s ) implies
a8).
&s
I
1 >,dN(s).
-
Proof. Necessity has been proved in Section 1.1 for condition H 1 , and i n Section 1.3 for condition H 2 with N = 1 , L = 3. W e suppose now that conditions H 1 , H 2 a r e fulfilled; a s each z. belongs to 3
6
HOL,OMORPHIC FUNCTIONS WITH GROWTH
IsN'
%($), t h e r e e x i s t s a positive i n t e g e r N ' s u c h that e a c h lz is uniformly bounJ ded. A s the constant 1 belongs to we get that 6 is uniformly bounded. Hence N and we may assume that N 2 N ' and t8 g 8 , W e take y =
aA),
Y(s)
1nf SEC"
=
(y(s') + l s ' - s l ) .
7
7
A s y,(fSN and )z16" is uniformly bounded, s a t i s f i e s condition \V 1 ; hence is a weight funchon. We also have ?<$. In the o t h e r direction, we u s e the following lem-
ma to prove that L,emma 1
7 is equivalent t o 8 .
.- k t 6,8' b e two non negative functions on
C" s u c h that
that for some positive integer N and some c > O , the relation S(s')>/d"(s);
then F>,e8N.
> Isl-sf
W e only have to u s e 6(sl)+i s ' - s f
N 8 ( s l ) + / s l - s ~> , S ( s ! ) i f ~ s ' - s l < c S ( s ) , to get
F(s') +
.
3 8 We assume N (s) implies
(s) and
lsl-sl&~8
Y
I s ' - s I 3 €XI' ( s )
on s '
in a l l c a s e s . Taking the infimum
N
if
,$I
IS'-S~<E~
, we have -8 3 ~ N8 ' .
A s an example, e v e r y non negative function such that @ ( s ) - S(S')l
6
on Cn satisfying W 1 o r H 1 and
,< c I s - S ' I
ci
with C 2 0, x > O , is equivalent to a weight function : i f N,E a r e chosen s u c h that N N > l / ~ ,C f " G - 5 a n d i f I s - s ' ~ < E ~ ( s ) , we have &sf)
>/
b)-
C&%(S)
>,
+
$(s)
and condition H 2 follows.
is
We s h a l l see that e v e r y weight function 16 > 01 More p r e c i s e l y
cmon
Proposition 4 .
-
.
6
is equivalent to a weight function which
- Let 8 be a weight function on Cn and E be a s t r i c t l y positive cons& , which is c " { ~ 8 > 0 ] , s u c h that (1 + E l 8 ( 1 - € ) 8 < 8'
tant. T h e r e exists a function and
/D8/
=
for e v e r y derivative D of order r
O((~E)~-~) 1.
Proof, W e may assume that 8 61, & 4 . For e v e r y non negative i n t e g e r p , let S P denote t h e s e t of all points s in Cn s u c h that 6(s) >, ( 1 E)p. W e c o n s i d e r a non negafunction 'p on C", with support in the unit ball i\zI ,c 11, s u c h that tive
em
-
7
REGULARIZATION O F WEIGHT F U N C T I O N S
and we define, for e v e r y p , a function
Let a l s o
1
jf
YP
by
be the c h a r a c t e r i s t i c function of S
P
and
(c (c
yp1< 1
As Cp ) d h ) = 1 for e v e r y p , we have 13( X C" P uniformly convergent. W e s h a l l f i r s t estimate Ft ; note that i f s belongs t o S 1s-S'I
6
E(l
P
and the series is
and s a t i s f i e s
-€P,
then s' belongs to SWl , b e c a u s e &s',
>,
(1
-
L)P
-
E ( l - E ) P = (1 -
X q * c ~ , ( s=) 1 for q 2 p + f , as i n the support of and Hence
yq,
IS-S"
,<
,< p - 2
-
by a similar argument, and
a(s)
G
2
E(i -6)q
=
.
( ( T - E ) ~ f o r e v e r y p o i n t s'
On the o t h e r hand, if s does not belong t o Sp 1 , we have
q
€ ) P+l
xq
1: 'Q ( S ) = 0
(1 - E ) P - ~ .
q3,P-l
A s the property is valid for e v e r y p , w e get
and t h e good estimate follows, when replacing & by &/2.
Let D be a derivative of o r d e r r 3 1. We have
and t h e series is locally uniformly convergent, because if
9
for
HOLOMORPHIC FUNCTIONS WITH GROWTH
8
and Dcpp d o not vanish a t s. Moreover only D '4 P- 1
1 .5. - Inductive limits In this section w e consider algebras of holomorphic functions which are inductive limits of al g eb r as introduced in Section 1.1. be a set of non negative functions on C " ; we assume that A is directed Let
8"
6 , E A , t h er e e x i s t s some 8 e h su ch that for some positive integer N and some E > 0 . By consideN in A such that E & 6 for some N , E , the r est r i ct i o n
in the following s e n s e : for e v e r y E
g N ,<
sf ,
r i n g , for
ES N g
5 , FI
mapping
st
%( Sl)
3
?JD
t
we define an inductive system. The d i r e c t limit of this system is an al g eb r a; i t is
denoted by &(A). Elements of Z ( h )are obtained by identifying in P A functions which are equal on some { s > O 1 If each > 0) is assumed to b e open, the d ir ect l i m i t of the system
cs
subalgebra of
.
z ( A ); i t is denoted by 0 (A).
We d i s cu s s a few examples of a lg e b r a s O(A), where functions or can be replaced by such a directed s e t ,
into
1) Let [O,af
a$)a is
A i s a d i r ect ed set of
weight
be a n open s e t in C". For e v e r y convex increasing mapping of (0, OOC
, we set
Lq
=
exp(-y(-log
sn,cp
Sa)).
Then b, denotes the set of all functions which are weight functions. It is easily seen that b is a directed s e t because the function Min ( 8 0 , , ~ q , ) is
n
sn,
INDUCTIVE LIMITS
9
a ss o ci at ed to Max( (p, rf') and belongs to A
n'
Proposition 4.- For e v e r y open set 62 of all holomorphic functions on
o(n)
n.
C", the algebra
o(A,) is the algebra
It is an eas y consequence of the following Lemma 2 . - For ev e r y increasing mapping 9 from [ O , a r to [ O , a [ , t h e r e e x i s t s a convex increasing mapping 'p from [O,mf to cO,m[ such that y<
is Lipschitz with constant 1
=
exp(-y(-logx))
0210, 11 .
Proof of lemma 2 , W e consider the sequence c = ct) ( n + 1) and define g by induction n on [ O , n] a s a convex increasing majorant of s u ch that
for ev er y y in [ O , n]
, where
g ' ( y ) is the left or ri g h t derivative at y , and g(n)
2 cn.
W e define g on { O ] by g(0) = c, and, i f g h a s been already defined on [ O , n j consider the unique solution defined on a right neighbourhood of n of dh
, we
h
F=e such that h(n)
=
g(n). Thus =
n
+
.-g(n)
- .-h(Y),
and h tends to infinity a t a point c of ] n , n+l[. Obviously h is convex increasing on = eg(n) 2, g ' (n), if we extend g by h , we obtain a convex inBy replacing h on [c' ,+ob[ by the affine function which c r eas i n g function on [ O , c
[n, c ( an d , a s h ' (n)
[.
is tangent to h at a point c ' of 3 n , c [, we obtain a convex increasing function on (0, +-[ and we may assume that its value at n + 1 is l a r g e r than c ~ +W~e take . this function as g on [ O , n+l] As g(n) 3 y ( n + l ) , and g , Y are both i n cr easi n g , we have g(y) 3 y ( y ) for e v e r y y in ] n , n+l] and the induction is complete. Let 'Q b e defined by
.
'Q(Y) = 1 + Y + dY). Obviously
'9
is a convex increasing majorant of
and as e-g(y)gl(y)
< 1 , we get
1 f ' (x)l< 1.
Y
. Moreover
Hence f is Lipschitz with constant 1.
HOLOMORPHIC FUNCTIONS WITH GROWTH
10
Proof of Proposition 4 . Let
Y
b e a convex i n c r e a s i n g mapping of
&
[O, m
s
into
is also (0,00 [ and 'p as in Lemma 2. A s is Lipschitz with constant 1 , QQ Lipschitz with constant 1 and t h e r e f o r e s a t i s f i e s condition W2. If we assume that
>
$a,+,<&
also s a t i s f i e s condition W 1 . W e only have
y , we get and y(y) t o prove that e v e r y f i n (3(n) s a t i s f i e s
e x p ( - ~ ( - l o g S ~ ) ) / f1 I ~
or log
If1 s
qJ ( - l o g La,,
into [O , go [ for some increasing mapping of (0, a0 compact i n fi for e v e r y Y E [0, oO[, we c a n take
2) Let K be a compact s e t in C". If
. As the s e t
AK denotes the s e t of
{& > e-']
is
all weight functions
where c1 is an open neighbourhood of K , t h e a l g e b r a 8 ( A , ) is the a l g e b r a of germs of functions which are holomorphic on a neighbourhood of K
.
&,
O(K)
3) The a l g e b r a of e n t i r e functions of finite o r d e r on Cn is t h e d i r e c t limit of the a l g e b r a s O(e-''lk), where k v a r i e s in N. Let u s more generally c o n s i d e r a non bounded convex i n c r e a s i n g mapping of
.
,
[o, m[
into lo, as[ called cp We denote by A the s e t of a l l functions
'9
hc
=
exp ( - y ( - l o g c
and c is a positive constant. C l e a r l y on Cn such that
If 1
=
1, where
(')(A+
So))
is the a l g e b r a of a l l e n t i r e functions f
O ( e C ~ ( - 1 o g cS o ) ) ,
so
so)
for some c > O , C 3 0 . A s l o g \ z ( S - l o g and - l o g ( f < log ( 2 1 for I z I >, 1 , it is easily s e e n that (3(Ad is t h e a l g e b r a of all e n t i r e functions f s u c h that
If\
=
O ( ec c q ( - l o g l z l + C V )
for some C ' > , O , C > O . W e can find constants
Therefore W1.
xc
coo, M >/ 0, s u c h that
= O ( [ z \ - ~ )at infinity and
-
(f(x) >, cix M for all x E [O, OOc is equivalent to some function satisfying
-
.
'xc is not equivalent t o some weight function. w," s h a l l see, however, that is equal to (3((lc)). It suffices t o show that 'xc is l a r g e r than some func= hc and $ =Acb tion equivalent t o Ac,2. W e u s e Lemma I of Section 1.4 with Maybe
o(A'p)
8
INIIUCTIVG LIMITS and r e m a r k that
1s-s'J
1/2
8,(s) implies
.
Ac
When
L-l is an
all functions
open s e t in C" and
hc w h e r e
1, = Now
o(h
'4
a,
&,(s')
3
1/2 &,(s)
and
is l a r g e r than some function equivalent
T o complete the proof, we remember that to
11
a s above, w e denote by
e x p ( - 'Q ( - l o g c
h
'4
the set of
Xn)).
) is the algebra of ail holomorphic functions f on
0
s u c h that
/ f j = o(e
for some c > 0, C >, 0. We similarly show that hc is equivalent t o some function satisfying W1 and that (3(h ) = 'p Thus A?) o r (3(A ) are equal t o a l g e b r a s where is a 'p d i r e c t e d set of height functions.
o((ic)).
o(
o(b.),
Proposition 1 of Section 1.3 can be generalized in the following way : Proposition 5 . - Assume that e v e r y
O(A) G o O(A). , A' 6' E A
L,et A
8~ A
s a t i s f i e s condition W 1 ;then, a / b z . sJ
be two directed s e t s of weight functions on C n . W e suppose that for for some positive i n t e g e r \ t h e r e e x i s t s & A s u c h that X I 2 E.
s
every 8') -+ gives rise t o an and some E > 0. T h e s e t of r e s t r i c t i o n mappings homomorphism A t )3 Z ( A ) . W e similarlj have a n homomorphism A')-+ (A); i f we also suppose that for e v e r y 6' E A' the s e t { 8' > 0) is connected, and that no
z(
z(
5~ h
z(8)
o(
0
is identically z e r o , i t can b e e a s i l y proved that this l a s t morphism is injective :
O(A9 can be
in s u c h a c a s e ,
Notes. -
considered a s a subalgebra of
o(A).
z(s),rr(s) o(s),
and and conditions W 1 , W 2 have been introduced Algebras by L. Waelbrceck ( I ) (2). Propositions 1 and 2 a r e s t a n d a r d ; for a different proof of Proposition 2 , see I . Cnop (2). Conditions H 1 , H 2 have been considered by L. H o r mander ( j ) , and J.J. Kelleher and B.A. T a y l o r (') : they denote by A(?) the a l g e b r a = e-'Q . Proposition 4 is s t r o n g e r than a similar r e s u l t proved by (3(s) when
F
L. Waelbroeck ( I ) and uses a method taken from a p a p e r of the author (2). A l g e b r a s of holomorphic functions of exponential type or of finite o r d e r a r e classical : see C . O . Kiselman ( I ) and A. Martineau ( I ) . Algebras (3(&$ have been considered by L . Rube1 and B. A. T a y l o r (') and denoted by E ( A ) w h e r e 1 is the function x
H
Cp(1og x ) .
CHAPTER I1
BOUNDEDNESS AND POLYNORMEDVECTOR S P A C E S
W e introduce a s t r u c t u r e on the a l g e b r a s of holomorphic functions with
growth w e have considered in C h a p t e r I . T h i s is not a topology but a boundedn e s s . We define f i r s t polynormed vector s p a c e s as vector s p a c e s equipped with a suitable covering by pseudonormed s p a c e s . W e introduce then bounded s e t s , convergent and Cauchy s e q u e n c e s , T h i s l e a d s t o the definition of Hausdorff and complete polynormed vector s p a c e s .
2.1.
- Polynormed vector
spaces
6
We consider along this section a non negative function on C" which is assumed to b e bounded; w e have defined i n Chapter I the a l g e b r a g(8) of &-tempered complexwith a valued functions on C". In the g e n e r a l case, t h e r e is no way t o equip norm. However
%(8)
as)
naturally c a r r i e s t h e family of pseudonorms
where N r a n g e s o v e r Z. It is e a s i l y s e e n that t h e s e pseudonorms are equivalent i f and only if
8
and 1 / &
a r e both bounded on the s e t S = (6,O) ; t h i s means that 8 is equivalent to t h e char a c t e r i s t i c function of S. In s u c h a case %( 8) is the Banach a l g e b r a of all bounded complex-valued functions on S. W e note that a weight function identically vanish is never a c h a r a c t e r i s t i c function, Returning t o the general c a s e , we have
6
which d o e s not
POLYNORMEDVECTORSPACES where
Nc(8) is the s p a c e of
13
all complex - valued functions f on
I8>0) s u c h that If 18N
is bounded, equipped with the norm
8
Each .%(g) is a Banach s p a c e ; as and the identity mapping
is bounded,
is continuous. It would b e possible t o consider on
.ax),
N%(8) is contained in N+,E(8)
G(8) the
d i r e c t limit locally convex
topology of the sequence but such a topology is not e a s y t o handle. We therefore introduce s t r u c t u r e s which are closer t o the estimates leading t o the definition of
E((6).Roughly speaking we do not take the limit but
p r e f e r work with the system itself.
By definition a polynormed v e c t o r s p a c e is a complex vector s p a c e E equipped with a covering (iE)icI by pseudonormed (*) vector s p a c e s s u c h that I is a directed o r d e r e d s e t and, for e v e r y i ,< j , the identity mapping is continuous f r o m . E into .E. J
L e t ( E , (iE)icI) and (F, ( . F ) . ) b e polynormed vector s p a c e s . W e s a y that a J JEJ l i n e a r mapping u of E into F is bounded i f for e v e r y i c I t h e r e e x i s t s j a J s u c h that u is a continuous mapping of . E into .F
.
J
L i s t a f e w examples : 1 ) T h e vector s p a c e F(8)equipped with the covering (N%(8)),,z the beginning of this section is a polynormed vector s p a c e .
considered in
2) When the s e t {6>0] is supposed to b e open, we can d o t h e same for considering for e v e r y N E Z the vector s p a c e
of all functions in
NE( 6). The couple
,z(s)
N
o(8)= N g ( 8 )
(7
o(8)by
o(8)
which are holomorphic, equipped with t h e norm induced by
(o(a), (,as)),z) ~
defines a polynormed vector s p a c e . Propo-
sition 1 of Chapter I may b e completed as following :
If 6 is a non negative function on C" satisfying condition W 2 , bounded l i n e a r mapping of o( 8) n i&
o(8).
each
a/az. is a J -
3) W e may also r e g a r d Gr(6) as a polynormed v e c t o r s p a c e ; for e v e r y N E Z , we t h e vector s p a c e of all functions f in s u c h that lDfl denote by Ncr(8)
cr($)
is bounded for e v e r y derivative D of o r d e r s < r , equipped with t h e norm f H Max
io(w
(sup
6(s)i0
SN-'%)
ID@f(s)l1,
SN-'
BOUNDEDNESS
14
where rn is a multi-index with length IDC\ and D" the derivative associated to covering satisfies the r e q u ir e d conditions.
(Ncr(8))NEZ
o(
. The
If 6' is another bounded non negative function on C" su ch that & > for some positive number & and some positive integer N , i t is clear that the r est r i ct i o n map-
dN
ping 'G(P)-+TX$) ( r e v . 0(61)+0(8), F(S!)+G,(F) if open) considered in Section 1.1 is a bounded l in e a r mapping. 4 ) When
h
@>oI , @ ! >01
are
is a directed s e t of bounded non negative functions on C", we equip
C(A)( r es p . O(A) when e a c h 16>0} is open) with a polynormed vector s p a c e ; we consider on couples
covering which makes it into a the relation
(s , N)E. A x 2
( 6 , N ) S ( 8 I , N 1 ) definedby I' 6" is bounded by a positive multiple of 8 I!, and for each ( 6 , N), we take the pseudonormed vector s p a c e which is the image of N
TAX)
( r es p .
,0(6))
in
UO).
5) Let = ( E , (iE)iGI) be a polynormed vector space. We introduce a new polynormed vector s p ac e %( 8 ; &) a s the union, where i v a r i e s in I and N in Z , of the
Nc(
pseudonormed vector s p a c e s 8; Ei) defined as following : , E ( 8 ;Ei) consists of al l functions f defined on jX>O] and with values in iE such that $N (s)f(s) is uniformly bounded for 8 ( s )> 0 .
Ncr(
Nrq($; q),
When 18>0] is open, we also define the subspaces $; Ei) ( r e s p . Ei)) of al l functions i n Ei) which are r - times differentiable ( r e s p .
$(s;
,,,g(g;
r - times continuously differentiable, holomorphic). Taking the union when N , i v a r y , we obtain polynormed vector s p a c e s denoted by "e,(8 ; E ) , E), $; E)
2.2.
gcr(8; o(
.
- Convex bounded s t r u c t u r e s
We may consider the category, the objects of which are pseudonormed vector sp aces and morphisms are bounded l in e a r mappings, The product of polynormed vector sp aces ( E , (iE)ieI) and (F, ( . F ) . ) is defined as the product s p a c e E x F equipped with the J JGJ covering (iE x .F) . . When F is a v e c t o r subspace of a polynormed vector J (1,~)eIxJ space ( E , (iE)ieI), we consider on F the induced covering ( F A .E)i,I, where F n i E is equipped with the pseudonorm of iE.
.
Two polynormed vector s p a c e s ( E , (iE)iGI), (E, (.F). ) are isomorphic above E J JEJ i f the identity mapping of E is a morphism in both d i r ect i o n s: for ev er y i e I , t h e r e
exists je J such that iE is contained in .F and h a s a f i n er pseudonorm, and inversely; J or 0(6),we obtain isomorphic polynormed spaces . A c l a s s of polynormed vector s p a c e s which are isomorphic above E is called a convex boundedness on E . Let ( E , (iE)ier) be a polynormed vector s p a c e . A su b set B of E is s a i d to b e bounded if B is contained in a n homothetic of the unit ball of some iE. It is easi l y seen
i f we replace Z by N when defining %(;(6)
15
CONVEXBOUNDED STRUCTURES that a l i n ear mapping u of a polynormed vector s p a c e into another is bounded if and only i f the image by u of e v e r y bounded s e t is bounded. A convex boundedness on a vector s p ace E is then uniquely determined by the s e t note first that % h a s the following p r o p e r t ie s :
.?I of
all bounded su b set s. W e
a) B , U B ~ E % if ~ ~ ~ B 5~ 3 $3. E , b) B ' c A
if
B'cB , B E % .
c) E is the union of In g en er al a s e t
44
99 .
of s u b s e ts of a given set E satisfying p r o p er t i es a ) , b), c ) is
.
called a boundedness or a bounded s t r u c t u r e on E Th e elements of % are called the bounded s e t s of the boundedness. If 9 is the boundedness of a polynormed v ect o r space E , we also have : CB) Every bounded set is contained in a n absolutely convex bounded s e t .
Conversely, if 9 is a boundedness on a vector sp ace E satisfying condition CB) , we may consider the o r d e r e d s e t & of a l l absolutely convex bounded s e t s of ?I and for ev er y B E 8 , the vector s p a c e EB spanned by B , equipped with the Minkowski functional
X>O
of B . We thus define a polynormed vector s p a c e (E,( E B ) B E E ) and it is easily checked
.
that the boundedness of such a polynormed vector s p a c e is Hence convex boundedn e s s e s on a vector s p a c e E a r e exactly boundednesses on E satisfying condition CB). Note that a subset A of
c($) ( r e s p . o(6))is bounded i f and only if t h e r e exist a
positive integer N and a positive number M such that B is contained in the s e t of al l complex-valued functions on {&>O) such that
Convex boundednesses a r e often used as tools in the study of topological v ect o r spaces . For instance, if E is a locally convex topological vector sp ace, the Mackey boundedness of E is the s e t of a l l s u b s e t s B of E su ch that for ev er y neighbourhood U of the origin t h e r e e x i s ts
some &>O with EB C U.
Proposition 4 of Chapter I is completed as following : Proposition 1
.- Let
Mackey boundedness of
Proof. A s u b s et B of
a n open s e t in C n ; the boundedness of
o(Ad is also the
an), when equipped with t h e compact open topology. O(n) is bounded for the compact open topology i f
for ev er y compact s e t K in
a t h e r e e x i s t s a constant MK
and only i f
which is a uniform bound on
16
BOUNDEDNESS
K for al l functions in B. From this follows that e v er y bounded set in ded in O(n). Conversely, if B is bounded in taking
o(n),
and a majorant and thereby in
of
4
O(An).
such that
8
n,cp
O(An) is boun-
is a weight function, B is bounded in
Note that the boundedness of a polynormed vector s p a c e
O(8
E is also determined
“J’f
)
by
the set of a l l pseudonormed vector subspaces N of E su ch that the identity N + E is a bounded l i n ear mapping, Such pseudonormed vector s p a c e s N will be called pseudo-
E.
normed vector s p ac e s of the definition of
2.3.
- Completeness
Let _E = ( E , ( .E)ieI) be a polynormed vector s p ace, we s a y that it is Hausdorff if each iE is a normed s p a c e ; this means that th e r e ex i st s no bounded l i n ear subspace
.
except 101 We s a y that i t is complete i f , up to isomorphism above E , each .E is a Banach s p ace; i t is equivalent to a s k that e v e r y bounded su b set of E is contained in an absolutely convex bounded subset B such that EB is a Banach sp ace. A sequence (x ) in E is said to converge to x E E ( r e s p . to be a Cauchy sequence) P i f t h er e e x i s t s some iE such that the property holds in iE. Th i s is equivalent to the existence of a bounded subset B and a sequence E tending to z e r o in C su ch that P
x
P
- X
E E B
P
(resp.
x
P+9
- x € E
P
P
B
for q a 0 ) . Note that ev er y convergent sequence h a s a unique limit in a polynormed vector space i f and only if i t is Hausdorff. If E is Hausdorff, a n ecessar y and sufficient condition for completeness is that for e v e r y i~ I we can find some j E I such that ev er y Cauchy sequence in .E is convergent in .E To prove the sufficiency, choose for eveJ r y ic1 some jc1 with the above property and such that iE is continuously mapped into ;E. We have morphisms
.
J
E
i
+
iE 3 .E.
J
Taking the image .F of iE in . E , we also have morphisms 1 J i
E
-+. F +
.E. J
Obviously, each iF is complete and (iF)ieI defines a polynormed vector s p a c e which
COMPLETENESS
17
is isomorphic to E .
Let (x ) be a sequence of E and ( A ) a sequence of positive numbers. We write P P x = O(xp) ( r e s p . xn = o(xp))i f t h e r e exists a bounded sequence (y ) ( r esp . a seP P quence (y ) tending to z e r o ) such that x = kp P P W e consider a bounded non negative function y[ on C". A s each ,%(&) is a Ba-
.
nach s p ace,
c(8)is complete.
Proposition 2.-
F
F u r t h e r , a s easily shown:
is lower semi-continuous, convergence in
G($)implies compact
ax),cr(8) are complete. o(A ) is not always complete when o(6) is complete for ev er y
convergence in
~S>O]
86 A .
For instance, i f is the open d i s c in the complex plane with cen t er at P E N and r ad i u s P = Uqap uq , as Snp is a weight function, we know that ) is comand P plete. However, if A is the s e t of all 8, , it is e asi l y checked that o ( A f i s not Hausdorff : choose a sequence & of posifive numbers tending to z e r o and define f P in by f(s)= E for s e w p ; replacing f by z e r o on w, u u o d o es not P P change the class in a n ) and th e r e f o r e f tends to z e r o in O(A).
a(
3,
...
o(h0)
An example of complete Proposition 7. - L e t tive functions on C " ;
A
O(A) is given b j
be a directed set of bounded lower semi-continuous non nega-
if 16>0) h a s finitely many connected components for e v e q
then O(A) is complete.
o(A)
SEA,
.as)
W e only have to prove that is hausdorff: each is a Banach s p a c e ; i f the image of .0(6) i n O(A) is a normed s p a c e , t hi s is a Banach space. By ab su r d , A and a assume that f tends to z e r o in O(A);this means that t h er e exist sequence (f ) tending to zero in such that f r e p r e s e n t s f for ev er y p. Let P P W be the union of all connected components of 18, > 0) which i n t er sect {8>0] for e ver y 8~ A . W e can choose some &E A such that d2N< 6 , for some positive inte-
o(&,)
. 4 on is2 > 01, Then (f ) is a constant sequence tending to zero in as,); thus f = 0 P P on > 0) and f 0. W e associate to e v e r y polynormed vector s p a c e ,E = ( E , (iF)icI) a complete poly-
ger N and some positive number t and that ia2 > 03 is contained i n &I. A s f and P f coincide on some 0) and are holomorphic on $8, > 0). , they coincide on a and =
normed vector s p ac e a s follows : for e a c h i E I , we denote by iEthe Banach s p a c e associated to iE and by iF the image of i$ in the diTect limit F of the system iE^. Obviously (F, (iF)icI) is complete; it is denoted by
&.
BOUNDEDNESS
18 2.4.
- Closure and density
Let (E, (iE)iGI) be a polynormed vector space. If A is a subset of E , we denote by A the set of l i m i t s in E of elements of A. W e say that A is closed if A = A. It is is not necessarily closed; the closure of A , defined as the important to note that smallest closed subset containing A , may require transfinitely many operations. Now consider a vector subspace F of E We remark that, by definition of limits in
.
E , w e have
F
=
UiEIci,1E ( F n i E ) ,
whereC1 (F n i E ) is the closure in iE of F niE, The covering (Cl, (FniE))ieI iE 1E enables u s to consider F a s a polynormed vector space. W e always have morphisms
F + F + E ,
F
but
is not in general isomorphic to a vector subspace of E.
A vector subspace F of E is called dense in E when the polynormed vector
and E a r e isomorphic, This means that for each k 1 , there exists jeJ such spaces that every element of iE is a limit of elements of F according to the pseudonorm of .F This implies in particular that elements of E a r e l i m i t s of elements of F. J If F is a vector subspace of a complete polynormed vector space E , it is easily
.
seen that
F = P ; therefore F
is complete.
2 . 5 . Algebras and ideals A polynormed algebra is a polynormed vector space fitted out with a structure of
algebra such that the multiplication is a bounded linear mapping, This means that the boundedness satisfies condition AB) The product of two bounded s e t s is bounded. A convex boundedness on an algebra A which satisfies condition AB) is called an
algebra boundedness
.
tr(&, 'G ( A ) or O(A)a r e polynormed
For instance, ?%I, (3(6), Gr(8), algebras. More precisely, the identity mapping
is continuous for all N , P E 2 and the similar properties a r e valid for
GCr(8). If
are. Let
is a polynormed algebra, also
A
=
( A , (iA)icI)
0(8),
cr(8),
%(8; k),o(6;9, %,(8; A), %cr(8; b)
be a polynormed algebra and A[X]
denote the algebra of
ALGEBRAS AND IDEALS
19
polynomials with coefficients in A. W e consider on A [XI the covering by all v e c t o r subspaces iA
+ iA.
X
+. . .+ i A . X N ,
identified with products iAN+', when i v a r i e s in I and N in N. T h u s ALX] is a polynormed a l g e b r a ; a subset B of ALX] is bounded if the d e g r e e s and coefficients of elements of B are.
...
W e define similariy A [XI, ,Xn]. For i n s t a n c e , the polynormed a l g e b r a s (3(&,) , ,Xn] are isomorphic ; t h i s is a consequence of Liouville s Theorem.
and C [X
.. .
A vector s p a c e E equipped with a complete convex boundedness (that is a c l a s s of complete polynormed vector s p a c e s which a r e isomorphic above E) is called a b-space. An a l g e b r a equipped with a complete a l g e b r a boundedness is called a b-algebra. W e have a l r e a d y found many examples of b-algebras; note that if A is a b-algebra, also
c(&; A) o r A [ x ~ ,. . . ,x,J are.
An ideal I of a commutative b-algebra A , equipped with a complete convex bounded-
n e s s , is said t o be a b-ideal i f both the identity mapping 1 + A and t h e multiplication A x 1 -+ I a r e bounded linear mappings. It is equivalent to a s k that e v e r y bounded s e t in I is bounded in A and that the product of a bounded s e t in A by a bounded s e t in I
is bounded in I . Let a,,
...,aP be elements of a commutative b-algebra I
generated by a , ,
=
idl(al,.,.,a
A.
We equip the ideal
.A) P'
.,.,a P with a s t r u c t u r e of b-ideal
a s following. Assume that the boundedness of A is associated with a covering (iA)iaI by Banach s p a c e s . W e consid e r the covering of 1 defined by iI
=
a l .iA+...
+ a p . iA,
where iI is identified with the quotient
.
iA x. .x iA/Ker (pi
of the product iA x.. .x iA by the k e r n e l of the linear mapping ' p i : (x
,,...,xP -+ a l x , + ...+ aP xP'
I t is e a s i l y seen that the identity mapping I +A
and t h e multiplication A x 1 + I
are
bounded l i n e a r mappings. From the f i r s t p r o p e r t y , we deduce that I is Hausdorff a s A is. Then, e a c h 11 is a Banach s p a c e and I is complete.
W e have to prove that the boundedness of I only depends on t h e boundedness of A . Note that a s e t B in I is bounded i f i t is the image by some iA x . . .x iA. T h i s means that t h e r e e x i s t bounded sets B1,.
B C a, B1 +.
. .+ aP BP'
Ti of
a bounded s e t of
. ., BP
in A s u c h that
BOUNDEDNESS
20
,
.
(of c o u r s e , we may choose B =. .= Bp). Similarly, when I , , ,In are b-ideals of a b-algebra A , t h e r e is a n a t u r a l way t o equip I I t.. .+ I n with a s t r u c t u r e of b-ideal. A s u b s e t B of I is bounded if t h e r e , ,In s u c h that B C B,+. . + B n . e x i s t bounded sets E l , . , E n in I , ,
.. . ..
.
..
Let I b e an ideal of a commutative Banach a l g e b r a A with unit element. Then 1 = A a s soon a s I is the limit of a sequence of I , because the s e t of invertible
c~lemcntsis a ncighbourhood of 1. The statement is no longer valid when A is a bHowever
a &bra
.
Propositioii 4 ( L . LLaelbroeck).-
Let I a-b-ideal
of a commutative b-algebra A w & h
or,
unit element. T h e n I = A if 1 is the limit in A of a bounded sequence in I , r n n generally, i f there e x i s t s a sequence (x ) s u c h that x = O ( k p ) 2 1 , 1 - x = O(k:) PP P i n A , uAh k l k 2 < I . I’roof. We only h a \ e to sholc that 1 belongs to I ; i f such a property holds, a s t h e multiplication by 1 gives a morphism A+ I , w e obtain t h e equality I = A between b-spaces. Setting y
I’
=
I -x
P’
arid v r i t i n g xp+l-xp
u e get x
P+l
- x1’
=
O((k, kZ)’)
=
YpXp+l
in 1. T h e r e f o r e
c
p>o
(x - x ) P+l p
=
- XpYp+l
2
Pa0
(x - x ) c o n v e r g e s in 1. But P+l P
I-x,
i n A ; hence 1 ~ 1 . llre a l s o need a more p r e c i s e r e s u l t . I of a commutative b-algebra A with unit Proposition 5.- We consider a b-1 J I , a normed s p a c e E of t h e definition element, an absolutely convex bounded set B U
-of
A and a Banach s p a c e F of the definition of I s u c h that EB a&
nuously mapped into F . the c l o s u r e of B 2 F .
E x EB are conti-
1 is the limit in E of a sequence of B , then 1 belongs to
Proof. We may assume that UyxllF ,< llxllE for all X E E , y t B. For e v e r y &>O, we chcose a sequence (x ) in B such that y = 1 - x s a t i s f i e s (Iy (I & 6 2 - P - 2 . Then P P P P E
and
c
P20
( x w l -xp) converges in F; t h e r e f o r e I E F and
NOTES
21
Notes. (*) A pseudonormed vector s p a c e is a v e c t o r s p a c e equipped with a
finite pseudo-
norm. Bounded s t r u c t u r e s have been f i r s t studied in a systematical way by L. Waelbroeck ('). T h e exposition is different h e r e because w e put the emphasis on the family of pseudonorms which defines t h e convergence. S u c h a point of view is fitted t o the examples and problems we s h a l l c o n s i d e r . Actually, the category of polynormed v e c t o r s p a c e s is only equivalent t o the category of vector s p a c e s equipped with convex boundedness. For bounded s t r u c t u r e s and t h e i r application to functional a n a l y s i s , the
r e a d e r is also r e f e r r e d t o C. Houzel ( 1), H. Buchwalter ( I ) , H. Hogbe-Nlend (') and L. Waelbroeck (4),( 5 ) . Although most of the a l g e b r a s we u s e a r e Hausdorff and e v e n complete, w e consider pseudonorms when defining polynormed v e c t o r s p a c e s , b e c a u s e a l g e b r a s O(A) and O(K) are not n e c e s s a r i l y Hausdorff. T h e boundedness on F when F is a vector s u b s p a c e h a s been used by the author in (2 ). It is a n improvement of the previous consideration of the c l o s u r e , i n view of approximation problems. More information about t h e limiting operations which lead t o the c l o s u r e of a s u b s p a c e is given by L. Waelbroeck ( 5 ),
Proposition 4 is called "Fundamental Lemma" by L. Waelbroeck ('), (7).
CHAPTER 111
SPECTRAL, THEORY O F b-ALGEBRAS
W e define the spectrum of one or several elements i n a commutative algebra A with unit element. The case of Banach algebras is first discussed and the elementary properties recalled. In the c a s e of b-algebras, the consideration of the algebraic spectrum is not sufficient. We define spectral s e t s and spectral functions. A subset S of C" is said to be spectral for a , , if there exists a bounded set B such that ( a l - s l ) B
+. . .+ (an-sn)B
..,,q,
contains 1 for all (sl,. , , ,sn) i n the complement of S . The concept of a spectral
function is a refinement of that of a spectral s e t . When A is a Banach algeb r a , a subset S of C" (resp. a non negative function 8 on C") is spectral
.. .
for a l , , a if and only i f it i s a neighbourhood of the algebraic joint spectrum (resp. is locally bounded from zero on the algebraic joint spectrum). W e prove, in the general c a s e , that every spectral function is larger
than some spectral function which is a weight function. For e v e r y weight function 8 , spectral for a l , , ,a,,, we construct a bounded linear mapping
o(&into A
..
...
..
which maps p ( z l , , z n ) onto p(a,, . , a n ) for every polynomial p. This is the holomorphic functional calculus. W e also introduce a b-ideal I of A , consider spectral functions modulo I , and cons-
f
f [a] from
H
truct an holomorphic functional calculus which is a mapping f r o m (3(8) into A/I. W e prove that, when aif bi modulo I , spectral functions modulo I for al, f [b,
...,an
al,.
. ., a
. ..
..
and b,, ,bn a r e the same and that f [ a l , . , a n ] and a r e equal in A/I W e also prove that 0 is never spectral for
, .. .,bn]
.
modulo I unless A = I .
SPECTRUM I N A BANACH A L G E B R A
23
3 . 1 . - Spectrum of elements in a Banach algebra W e s h a l l only consider commutatlve a l g e b r a s A with unit element. T h i s assumption w i l l not b e explicitly mentioned. Most of the r e s u l t s remain however valid when A is not commutative, for elements taken in the c e n t e r of A.
T h e spectrum of a n element a of A is the s e t of all complex numbers s s u c h that a-s h a s no i n v e r s e . I t is denoted by s p ( a ; A) or s p ( a ) . More generally, l e t a , , , a n b c elements of A . The Joint spectrum of a l , . ,an is the s e t of all s = ( s , , ,sn) in Cn s u c h that t h e ideal
. .. ...
..
id1 ( a , - s l
..
,..., an - s n ' A ) , *
generated by a l - s l , . , a - s in A is different f r o m A . I t is denoted by n n s p ( a l , . . . , a n ; A) o r s p ( a l , . , a n ) .
..
W e now consider a Banach a l g e b r a , that is an algebra .I with a Banach norm s u c h that
.
11x0 IlY il ,
11 XY I
for e v e r y x , y in A . It is well known and e a s i l y shown that the set of invertible elements is a n open neighbourhood of t h e origin and that the mapping x F? x-l is continuous and even analytic. Proposition 1
.- Let a l , . . ., a n
. ..
b e elements of a Banach a l g e b r a A .
a ) T h e spectrum s p ( a l , , a n ; A) is a compact s u b s e t of C". b) We can find mappings u l , . , u n defined on the complement of s p ( a l ,
e"
..
and taking t h e i r values in A s u c h that ui(s)
=
O( Is\-') at infinity for i=l ,
. . .,a) . . . , n and
. .+ ( a n - s n ) un(s) = 1 , ( s , , . . . ,sn) in the complement of s p ( a l , . . . ,an). ( a l - s r ) u l ( s ) +.
for e v e r y
-
Proof. It is e a s i l y s e e n that t h e r e e x i s t s a n open neighbourhood V ( cL3) of infinity s u c h that ( a l - s l ) s l +.. .+ (an-sn)sn is invertible for s in V(00). Setting
7
w. (s) 1,m
Thus sp(al ,
Now fix (tl ,
=
- +. ..+ (an-sn)Fn)- 1 ,
Si((al-sl)sl
. ..,an) is contained in the complement of V and t h e r e f o r e bounded. ...,tn) i n the complement of sp(a.,, .. .,an) and choose elements of A (00)
such that (al-t,)vl,t
+. . .+ ( a n - t n ) v n , t
= 1
.
SPECTRALTHEORY
24
A s the s e t of invertible elements is open, t h e r e e x i s t s an open neighbourhood V(t) of t such that (a.,-sl)vl,t +. .+ (a,-s ) v n n , t is invertible for s in V(t). Setting now
.
Then V(t) is contained in the complement of s p ( a , , is proved.
Choose now a
ernpartition of unit
(V (t)) of the complement of s p ( a l ,
cpm,
(Yt)
. , .,an) . T h e r e f o r e
property a)
subordinated to the covering V(co), ?As) = 1 on a neighbourhood of
... ,afl) s u c h that
infinity. Obviously, each ui = Y m w i , a is e m a n d ( a l - s , ) u.,(s)
+
T y t
+. . .+ (a,-s
Wi,t
n ) un(s)
=
I.
Moreover, in a neighbourhood of infinity, we have Ui(S)
=
w.
1900
( s ) = o(lsl-l),
and the proof of b) is complete. A well-known property is the fact that s p ( a , ,
...,afl) is never empty.
When n
=
1,
this follows f r o m Liouville ‘ s Theorem as t h e resolvent function
s e (a-s)-l is analytic on the complement of s p ( a ) in t h e Riemann s p h e r e , W e s h a l l give a proof of the property in a more g e n e r a l setting a t the end of the C h a p t e r . T h i s c a n also b e
deduced from the consideration of the s e t M of all maximal ideals of A . We identify M with the set of multiplicative l i n e a r forms which d o not identicall y vanish. T h e kernel of a multiplicative l i n e a r form # 0 is a maximal i d e a l . Conv e r s e l y , i f m is a maximal ideal, m d o e s not i n t e r s e c t the set of invertible elements and is t h e r e f o r e closed. T h e quotient s p a c e A/m is a Banach algebra and a field. Thm A/m = C because e v e r y element which does not lie in C h a s an empty spectrum, and m is the kernel of the multiplicative l i n e a r f o r m A +A/m
=
c.
As usual M is equipped with the weakest topology s u c h that the mapping
3 : r-)!(a) is continuous for e v e r y a E A. T h i s identifies M with a closed s u b s e t of the product space
SPECTRAL S E T S
25
and therefore M is a compact s p a c e . Proposition 2 . sp(al,.
. . , an)
.
Let a l ,. , , a be elements of a Banach algebra A. T is the set of elements (?(al),
when X -
. . .,
2
%(an)),
ranges o v e r M .
It is obvious that e v e r y ( $ ( a l ) , . . . , % ( a n ) ) belongs to M . Conversely, i f id1 ( a, - s l , . . , a - s . A ) is different from A , it is contained in a maximal ideal m .
.
n
n’
1 b e the multiplicative l in e a r form associated to m. W e have i = l , . . ., n and then ( s,,.. ., sn) = ( % ( a l ) , . .. , %( an ) ) . Let
l(al-sl)
A s M is not empty, w e deduce f r o m Proposition 2 that nor s p ( a l , .
3.2.
- S p ect r al
=
. . ,a
0 for
) is.
sets
S p ect r a of elements in b-algebras a r e not n e c essar i l y compact. For instance, a s is the algebra of polynomials, we have s p (z; (3( = c ; if D is the unit open
o(8,)
d i s c in the complex plane, the spectrum of z in
o(8,)
8,))
is the unit d i s c itself.
W e f i r s t consider the spectrum of one element a of a b-algebra A . W e cannot prove
nice properties for the resolvent function s + (a - s)-’ on the complement of s p (a) and have to s e t a new definition. Definition 1
.- A subset
S
of C
is said to be s p e c tr al €or a
1”A, If (a - s) - l
e
s
and is bounded when s r a n g e s o v e r t h e complement of S. The s e t of a l l s p e c tr a l s u b s e ts S is denoted by U(a; A) o r 6 ( a) . Proposition 3.- The interior of e v e r y s p e c tr a l set for a is s p e c t r a l for a ; the resol-
vent s e (a - s)-’ is holomorphic in the exterior of ev er y sp ect r al s e t for a . Proof. L et S 6 ( a ) . W e can find a bounded set B in A such that (a-s)-’ exists and E
belongs to B for e v e r y s off S. If s is on the boundary of S , this is the l i m i t of a sequence (s ) of the complement of S. We have P
.
and i f E is a Banach s p a c e of the definition of A s u ch that €3 and B B are bounded i n E, obviously ( a - s )-I is a Cauchy sequence in E . Th er ef o r e ( a - s )-I h as a P P limit x i n E such that ( a - s ) x = 1 in A, and a - s is invertible. Moreover, when s
SPECTRALTHEORY
26
ranges over the boundary of S, it i s clear that (a-s)-' remains in a bounded subset of E . Thus the interior 3 of S also belongs to 6 (a ). Let u s consider now an interior point s of the complementof S.For s close enough to so, ( a - s ) - l exists and belongs to B. Using ( a- s ) - l
- (a-s0)-'
=
( s - s o ) ( a - s ) -? (a -so)- 1
[s
-
,
w e see that s H(a s)-' is continuous from into E . A s A is a b-algebra, 1 (s, t) H ( a - s)- (a-t)-' is also continuous from csx into some Banach space F of the definition of A such that the identity mapping is continuous from E into F Then 1 the resolvent function s *(a - s)- is a complex differentiable mapping taking i t s values i n F, and i t s derivative at so is equal t o (a - so)2;it is even continuously differeninto F tiable a s a mapping of
[s
.
.
[s
It follows from L,iouville's Theorem and the second part of Proposition
that p5
is never spectral for a . Hence a ( a ) is a t r u e filter i n the complex plane, the inter-
section of which is s p (a). Moreover a ( a ) has a basis of open s e t s . When A is a Banach algebra, d ( a) consists of all neighbourhoods of s p (a) : the resolvent function is bounded on the complement of e v e r y neighbourhood of s p (a) and conversely, for every S ~ d ( a )the , interior of S belongs to b ( a ) and therefore contains s p ( a ) . This is not valid for b-algebras; i n that c a se , d ( a ) gives much more information than s p (a), We now define the joint spectrum of elements a l ,
.. . , an
of a b-algebra A.
..
Definition 2.- A subset S 2 Cn is said to be spectral for a l , , , a n i f one can associate to every s = ( s , , , , sn) in the complement of S , elements u,(s), , un(s) bounded independently of s $ A such that
..
. ..
-
(al s 1) u 1( s ) +.
..+ (an- sn)u n ( s )
= 1.
. ..
...
The se t of all spectral subsets for a l , , a is denoted by a ( a , , , an; A) or b ( a l , . , , an). We shall prove at the end of the Chapter and in a more general setting that @ never belongs to U(al t . ? an) Thus U ( a l , , , an) is a true filter in Cn,
.
.-
. .. We note that S is spectral for a l , . .., an if there exists a bounded se t B such that 1 belongs to ( a, - s l ) B +. . .+ (an-sn)B for every (s,, . . . , sn) in the complement
of S. This condition can be weakened a s follows
. ..
Proposition 4.- In o r d er that a subset S of Cn is spectral for a , , , a n , i t suffices that there exist a bounded set B and a normed space E of the definition of A such that 1 belongs to the closure in E of [(a,-s,)B +. ,+ (an-sn)B] n E for every
-
.
SPECTRAL S E T S
. ., s
(sl,.
27
) i n the complement of S.
..
Proof. F i r s t
fix s = ( s l , . , sn) in [ S . Our assumption shows that 1 is the limit in A of a sequence of (a,- s l ) B +. .+ (a - s )B, that is a bounded sequence of the n n ideal id1 ( a , - s l , . , an- sn;A). It follows then f r o m Proposition 4 of Chapter I1 that
.
..
1 belongs to such an ideal; hence there exist elements u l ( s ) , that
(a1- s 1) u1( s )+.
. .+ (un-sn)
un(s)
.. ., un(s) i n A such
= 1.
..
The proof will be complete i f we s h o w that ul(s), . un(s) can be chosen i n a bounded s e t independent of S. Let F be a Ranach space of the definition of A such that E
and E x EB a r e continuously mapped into F.Let
B
B1
=
(al-sl)B
+...+ (an-sn)B
9
.
be theBanach space (al- s l ) F +. .+ (an- sn)F equipped with the norm and let considered i n Section 2 . Obviously EB., and E x EB a r e continuously mapped into 1 F1. i t follows f r o m Proposition 5 of Chapter II that 1 belongs to the closure of B1 i n Fl Then i f C is the unit ball of F ,
.
1
In other words
E
+ ( a l - s l ) C +. . .+
B,
1 E ( a l - s , ) ( B u C ) +.
(an-sn)C.
. .+ ( a n - s n ) ( B u C )
and the statement is proved a s B u C is independent of s . Proposition 5 . - The interior of every spectral set for a l , al,
. . . ,a n .
Proof. Let S E < ( a l ,
.. .,a n
is spectral f o r
... , an)
and choose coefficients u,(s) satisfying -sn)u ( s ) = 1 and contained i n an absolutely convex bounded set B. Every point s =' ( s l , , sn) of is the limit of a sequence t = P of Writing (tl , p , . , ( a l - s l ) u l ( s ) +.
..+ (a
r..
IS.
..
[s
( a l - s ) u (t )+ ...+( a n - s n ) u (t 1 - 1 1 1 P n P
=
(t
- s ) u (t ) + . . . + ( t n , p - ~ n )(t~ ),
'JP
n P
P
w e see that 1 belongs to the closure of ( a l - s l ) B +. . .+ ( a n - s n ) B in EB and Proposition 4 shows that 3 is spectral for a l , . . , a n . If A is a Banach algebra, a subset S of C" is spectral for a l , , a n if and only if it is a neighbourhmd of s p ( a f , . , an) : if S belongs to b ( a l , . , a,), also ? belongs to U(al , , , an) and contains s p ( a l , , a n ) ; conversely if S is a neigha ), Proposition 1 shows that S is spectral for a , , ,an. bourhood of s p ( a l , .
.
..
..
..
n
. ..
. .. ..
.. .
SPECTRALTHEORY
28
3.3.
- Spectral functions
In the study of algebras of entire functions for instance, the consideration of spectral s e t s gives no information on the algebra. The joint spectrum of the coordinate functions is always C". W e shall therefore need the following generalization of the spectrum.
Let a l , ... , a n
Definition 3.-
be elements of a b-algebra A. A non negative function
0" C" is said to be spectral for a i f elements u,(s), u , ( s ) , associated to every s = (sl,. . . , sn) 5 C", s o t ( a l - s l ) u , ( s ) +...+ (an-sn)un(s)
(3.3.1)
and u,(s), . . , , u n ( s ) -
+
..., un(s)
a r e bounded in A independently of s .
.. . ,a n ) .
8
can be
6 ( s ) u , ( s ) = 1,
The set of all spectral functions for a is denoted by A ( a , , Na,,
of A
...
. .. , a n ; A)
or
.
, a n ) : choose uo(s) = 1 and u l ( s ) =. .= un(s) = 0. A Obviously, I E A ( a , , n spectral function 6 gives some information a t points s in C such that g ( s ) = 0, or such that 8 decreases more or l e s s rapidly near s.
- Let a l , .. . , a So E A ( a l , . . . , an)
Proposition 6 .
be elements of b-algebra A ,
a) belong to A ( a l , . . , a,,), also M i n ( 8 , 8 I ) belongs to A b , , . , an) b) g $ belongs to A(a,, , an) E d for some positive integer N c) some positive number E , t h e n 8 belongs to A ( a , , , an).
8,
. ..
Proof. a ) Set -
ujs)
=
and
-zi
.
..
glZ~SN . ..
so(,) 2
for i = 1 ,
..., n ,
+. . .+ an;,, + 1) F,(s).
b(s) =
. . . , un(s) a r e bounded independently of s ( a l - s 1 ) u 1 ( s )+. . .+ (an-sn) un(s) + J,(s) uo(s) =
Obviously uo(s), u,(s),
..
and 1.
8').
u i , ...
b) Let u,, u , , , , un (resp. u;, , u;) be associated to $ (resp. We set u!l(s) = u.(s) for i = 0 , 1 , . , n i f &s)< $ ( s ) and u!l(s) = ui(s) for i = 0, 1 , . ,n 1 if
81&)< 8(s)t
..
..
Coefficients u&s) a r e bounded in A independently of s and satisfy
( a l - s l ) ul/(s)+. ..+(a,-s,)
u$s)
+
M i n ( s ( s ) , 6 ' ( s ) )Gb) = 1 .
c) We f i r s t prove that if 6EA(a,, . . . , a n ) and
gl& E 8
for some positive number
SPECTRAL FUNCTIONS
29
, then F I E A ( a , , . . . , a n ) . If u,, u l , . . . , un a r e a s s o c i a t e d t o 6 , we keep . . , un and take ud(s) = 0 i f d(s) = 0 and u;(s) = S S u,(s) i f $ ( s ) , 0. Coefficients u;, u , , . . . , u easily satisfy t h e r e q u i r e d conditions for 8' . N W e only have to :how that 8 E A ( a l , . . . , a n ) if 8 ~ A ( a , ., . . , an) and N is a po-
#
E
ul,.
s i t i v e i n t e g e r . Thanks to the p r o p e r t i e s already p r o v e d , we may assume that bounded. Taking (al-sl) ul(s)
+. . .+ (an-sn)un(s) +
8
is
8 ( s ) uo(s) = 1
a t t h e N th power, we g e t
Us) + 8N ( s ) u,N ( s ) =
1,
where U ( s ) is obviously bounded independently of s in id1 ( a l - s l ,
.. . ,a
- s n ; A).
It follows from Proposition 6 that e v e r y non negative function is s p e c t r a l a s soon a s i t is equivalent to a s p e c t r a l function. A s u b s e t A , of
function
6
in A ( a l ,
A(a,, . . .,an) IS s a i d to be a basis of A ( a l , . . . , a n ) if e v e r y . . . , a n ) is l a r g e r than some function in A,.
Proposition 7 . - A b a s i s of a ( a l , . . . , an) c o n s i s t s of a l l functions 'QB, w h e r e B a n absolutely convex bounded s e t in A a x Cq,(s) t h e distance in E B from 1 t_o (al-s,)B
+. . .+ (a,-s,)B,
and s u c h functions are Lipschitz o v e r C".
Proof. It is easily s e e n that if
6 is s p e c t r a l for a , , . . . , a n , t h e r e e x i s t s some absolutely convex bounded s e t B s u c h that 8 >/ yB; w e only have t o choose B l a r g e enough so that it contains uo(s), u l ( s ) , . . . , u n ( s ) . F u r t h e r , each yB is s p e c t r a l for al ,, . . , an. L e t S denote t h e s e t w h e r e yB
. .+(a,-s,)B in E B contains 1 . I t follows then from Proposition 4 that S is s p e c t r a l for a l , . , a n . When 'pB(s)= 0, we thus can find coefficients uo(s), . . . , un(s) which are bounded independently of s and satisfy
d o e s not vanish. For e v e r y point s $ S , the c l o s u r e of ( a , - s l ) B + .
(a1-s1) ul(s)
+. . .+ (an-sn)
un(s)
=
..
1.
When yB(s)70,i t follows immediately f r o m the definition of 'Qe(s) that t h e r e e x i s t s some u,(s)E B s u c h that 1 + 2
. . . , un(s)
in B s u c h that
( a l - s 1) u 1(s)+...+ ( a n - s n ) u n ( s ) + ( f g ( s ) . ZU,(S)
=
1.
SPECTRAI, THEORY
30
Now let s , s'be different points in Cn and
&
elements uo(s), u ](s),. . . , u n ( s ) of B s u c h that
a positive number. T h e r e e x i s t
.
u,(s) ( a l - s l ) u l ( s ) + . . + ( a n - s n ) un(s) + ( 'QB(s)+&)
=
1.
Then, we a l s o have (a
- 7 1' ) u 1(5)+ , . .+ (an- s;) ~
1
+
+
~ ( 5(cQB(s) ) E. ) u,(s)
+ (s - s ,) u , (s)+. . .+ ( s ; ~ -sn)un(s) =
1,
and (a - s l )u (s)+.. .+ (a -s')u (s) - 1 belongs to the s e t 1 1 1 n n n ( y B ( S ) + & 1+~ i - S , 1
Hence
cp,(s)+ E
(fB(S')<
t...+lS,',-S,I)B.
+ Jsi-sll +...+ Is,',+ I
a n d , a s E is a r b i t r a r y 'fB(") which shows that Corollary 1 .- g
,< (QB(')
(pR is 1,ipschitz o v e r
F
. Is;,-'"\
+. .+
+ \'\-'1\
I
C".
is s p e c t r a l for a l , . . . , a n ,
also g1-S
T h e r e e x i s t s some absolutely convex bounded s e t B sucJ that 1 fore $>,(p,. 1 As 1 s a t i s f i e s condition W 2 , w e have 8>;lq, for a l ,
..., a n '
8
..
If IS s p e c t r a l for a l , . , a n , also Min always a b a s i s of weight functions. It is obvious
( 8 ,so) is. T h e r e f o r e h
that for e v e r y s p e c t r a l function
6 , the s e t
and
and there-
IS s p e c t r a l
A(a, , . . . , an) h a s
{8>0) is s p e c t r a l .
Conversely, i f S is a s p e c t r a l set, the c h a r a c t e r i s t i c function Thcreforc
8 2, 'pB
Ys
is s p e c t r a l .
*
S = Min(XS,so)
is a l s o a s p e c t r a l function.
3.4.
- The holomorphic functional calculus
.. .
We consider a b-algebra A and elements a,, , , an i n A. W e s h a l l w r i t e (al a n ) = a , A(a an) = A(a) and x,y, +...+ xnyn = ( x , y > , when xn) and y = ( y , , , , y,) belong t o A". x = (x be a weight function in A ( a ) . O u r aim is t o c o n s t r u c t a bounded l i n e a r Let mapping of s(6)into A which maps p(z) onto p(a) for e v e r y polynomial p ; t h i s will
,...,
,""' F
,,...,
..
THE HOLOMORPHIC FUNCTIONAL CALCULUS
31
b e t h e holomorphic functional calculus a t a . W e f i r s t have to r e g u l a r i z e t h e coefficients which a p p e a r in the definition of t h e
spectrum. For e v e r y positive i n t e g e r N , w e denote by SN(a: 6 : A ) or S N ( a ; g) the
..
set of a l l continuously differentiable systems u = ( u l , . , un) defined on Cn and taking t h e i r values in A, bounded along with t h e i r d e r i v a t i v e s of o r d e r 1 and s u c h that y = 1
(3.4.1)
belongs to
Lemma 1
-
(a-z,
u>
-NT5 (
;A),
.- We have
SN+l(a;
8)C
SN(a;
8)
and e a c h S N ( a ;
J)
is non void.
Proof. The f i r s t p a r t of the statement is obvious and w e prove the second one f o r N l a r g e enough. It follows from Proposition of C h a p t e r I that t h e r e e x i s t s a weight 4
<
{s>O).
function 8I which s a t i s f i e s +6,(61 $8 and is e m o n the open s e t S = It is easily shown that &IN is continuously differentiable on C" f o r N 3 2 : for e v e r y N ) = 0 on the boundary of derivative D of o r d e r I , a s 6l is Lipschitz, c l e a r l y D( S and D(6IN) = N D J 1 tends to z e r o a t the boundary of S. Moreover
8'
SN-'
-,z,($) and PNc -NT(s). T h e r e f o r e the statement is an e a s y consequence of
y a continuously differentiable weight function i n A ( a ; A ) ; t h e r e e x i s t continuously differentiable functions u,, u, , . . . , u defined on Cn and taking t h e i r Lemma 2 . -
Let
values in A, bounded along with t h e i r derivatives of o r d e r 1 and s u c h that (a-z,
u> t
YIJ,
= 1.
Proof. W e a l r e a d y know that t h e r e e x i s t bounded functions (a-z,
v)
Choose a e m n o n negative function
+ Yv,
v,, v l , .
. . , vn
s u c h that
= 1.
P on Cn with support in the unit s q u a r e
Pql< 1 , ..., I X n l < 1 , ( Y , l < l ,
...
I
(Yn(
and positive on the s q u a r e
lx,l6 Let T = (2+ i Z )" and
Obviously
'Q
3,. . ., [X,I-C+,
yes, =
(Y,[&.
c
. ., [Ynl<3.
p)/ yJ(Sft). t ET
h a s the p r o p e r t i e s a l r e a d y mentioned and
positive i n t e g e r p and i = O , .
. ., n ,
we s e t
tET
'Q(s+t) = 1 , For e v e r y
SPECTRAL THEORY
32
is e m o n Cn and is bounded independently of p, whereas Clearly each w. 1, P = 0 (ZP) for every derivative D of o r d e r 1. Moreover L) w. 1, P
< a-s,
(3.4.2)
wp(s)
>
+ T(s)mr0, p(s)+
k (s) = 1
P
with k (s)
=
teT
Y(2's-t)
(2-'t-S,
+
V(2-'t))
t ET
'p(2Ps-t)(Y(S)
- 7(2"t))
VO(2-'t).
1 < 2-pi-1 G;thereby jy(s)- y(2-'t))
When ~ ( 2 ~t) s# -0 , w e have 1s- 2-'t
=
o(2-4.
Then, as easily shown, k (s)= 0(2-'). Taking now (3.4.2) a t the 4th power and P arranging t er m s , w e obtain coefficients CVOlp, W1 , p , , , W n , p such that
(a-z, W
P
..
>
+
+
YW,,~
kp 4 = 1.
is bounded independently of p , w h e r e a s k4 = 0 ( 2 - " ) . Besi d es Each lVi P tP = O(2') and D (k4 ) = 0 (2-") for e v e r y derivative D of o r d e r 1 . We may D \V. 1, P P apply Proposition 4 of Chapter I1 to the ideal generated by al- zl, , , , an- zn, in
cel(so; A):
w e can find w,, w l , . (a-z,
w)
The new coefficients w,, w l , .
. . , wn
in
Eel(;,;A)
.
Y
such that
+ T w o = 1. .. , wn belong to some N z C , ( l , ; A ) ;
they have
polynomial growth at infinity. In o r d e r to obtain bounded coefficients, we consider the coefficients
u1
=
-zl g,. . . , un = -zn so 2
and Y
=
(
+I)
-lce l ( z 0 ; C) and satisfy
of Proposition 6 a ) . They lie in
+ Y T hereforc
I
=
(a-z,
= (a-z,
Now U 1 + Y w l ,
U>
=
+ Y(
U + Yw>
..., U n + Y w n
So2
I.
(a-z,
w
> + yw,)
+ ~Yw,.
and Yw, l i e i n N - l ~ ~ l ( ~ o ; B Ay)d e. c r e a s i n g
..
induction on N , we easily obtain coefficients u,, u l , . , u n in -1 coefficients are bounded along with t h e ir derivatives of o r d e r 1 .
W e introduce now a differential form; writing dzl A . ~ d " u , = d"u,
d"ul
..
s)
A
. . . Adzfl = dz
(8,; A);
such
and
Proposition 8.- Let u t S N ( a ; and f E $(6); then f d"u A d z , extended by zero n on the complement of the s e t S = I s > O } , is continuous and integrable o v e r C for N >/ P + 2 n + 2 .
THE HOLOMORPHICFUNCTIONALCALCULUS
Proof. We f i r s t show that -
f d"u tends to z e r o a t t h e boundary of S. Using ( 3 . 4 . 1 ) w e
have
d"u
=
( (a-z,
and
d"u
=
33
( a 1- z 1 )u1 d"u 1 A
u
> + y) d"u
. . . ~ d " u+.. . + d " u l A . ,
. r\(an - z n )u n d"un + y d " u .
But, differentiating ( 3 . 4 . 1 ) , we get
( a1- z 1) d " u , + . . .+(a,,-zn)d"ul, + d"y
(3.4.3)
Therefore d"u
= -U
l d"y
A
=
0.
. . . A d ' u -. . .- d"ul A . . . A ( - ~ , d " y )+ y d " u ,
and, as y and the coefficients of d"y a r e in N-lT(S; A ) , also t h e coefficients of d"u A ) and fd"u tends t o a r e . When N > P + 2 , the coefficients of fd"u a r e in
-,z($;
zero a t the boundary of S. Extended by 0 on the complement of S , c l e a r l y fd"u is continuous. When N 2 I'+ 2 n + 2 , the coefficients of f d"u a r e in 42n-1) c ( $ ; A ) . A s $ is a weight function, f d"u = O( \ z I - ~ ~ - ~ )at infinity and is integrable o v e r C" Proposition 9.- L e t f
g
€[,a$). T h e element
A d o e s not depend on the p a r t i c u l a r choice of u
&
SN(a;
8) with -
N)P+2n+2.
W e f i r s t need
Let
Lemma 3.u, U ' E S N ( a ; 8 ) ; t h e r e existsacontinuous f o r m v of type ( 0 , n - 1) with coefficients in A ) such that d " u ' - d " u = d " v .
-,z($;
Note that v is not assumed t o b e differentiable : d"v e x i s t s in t h e distributional s e n s e , but the coefficients of d"v a r e continuous functions b e c a u s e those of d"u, d"uT are.
Proof. W e s h a l l prove the statement when n -
=
2 ; the g e n e r a l case is quite s i m i l a r , the
calculations being more complicated. W e have
i
(a,-zl)u,
+ (a2-z2)u2 + y
=
1
( a , - z1 ) u1l + ( a2 - z2) u 2v + y t = 1
where y , y ' a r e in some
-N%(s;
A). Therefore
SPECTRAL, THEORY
u ; - u I = [ ( a l - z 1)u 1 + ( a2 - z 2 )u 2 +y]u; =
-
[(a,-z,)ui
+(a2-z2)u;+y1]u1
(a 2- z 2 )(u2 u1t - u $ u l ) + y u ; - y t u l .
Calculating s i m i l a r for u $ - u 2 and y ' - y and setting
w e gct
i
ui-u, = (aL-z2)w+ uJ-u2 = - ( a l - z , ) a +
Y 1- Y
A s easily s e e n ,
-N
1 (8;A).
o(
=
-(al- z l )
GI-
t1
t2
(a2- z 2 ) t 2
is bounded along with its derivativeswhereas
F i r s t , consider the c a s e when
t1= E 2 = O ;
then
d"u'- d"u = (d"u 1+ ( a2 - z 2)d"c&,(d"u2- ( a , - z l ) d % ) = =
- (a2 - z 2 )d"u2]
[-(al-z1)dl1u,
A
t1,c2
a r e in
- d"u1Adt4u2
d"u
d"y A d"U
Thus d"u' - d"u = dl'v , i f v = y d " a
. Supposing now that
d " u ' - d " u = ( d " u l +d"( ) A d"u -d"u
1
1
= d"{, A d"u2
2
.
a=0 ,
t2 0 , we have =
1 A d"u '2
Thus d " u ' - d " u = d"v, if v = $d"u2. T h e case when N = O , $ - 0 , is similar.' To 1obtain the g e n e r a l c a s e , w e only have to decompose the transformation which, s t a r t i n g
from u , , leads to u 2 , through (ul +
G,, u2)
and (u,
+{, , u2+ E2).
Proof of Proposition 9.- Keep the notations of Lemma 3; we also have f d"u'A dz
- f d"u A
dz = f d"v
A
dz
= d(fv A dz).
Applying Stokes'formula, a s the coefficients of fv A dz are O ( ( Z ~ - ~ a~t infinity, - ~ ) we get
r
C" and the statement is proved.
d(fvhdz) = 0 ,
35
THE HOLOMORPHlC FUNCTIONAL CALCULUS Note that we only need the fact that fvr\ dz
=
O( \ z \ - ~ " )a t infinity to prove that the
integral of fv A d z on i n c r e a s i n g s p h e r e s with c e n t e r at 0 tends to z e r o . T h u s , i f fd"uA d z
lim
Izl
i1,,$2
and some € 7 0 , w e have SQN(a;$)C S N ( a ; 8 ) . If f is in
F or 6 , , is t h e r e f o r e the same.
calculated with
Proposition 10.- For e v e r y polynomial p , w e have p[a]
=
o(s)
O(x1),t h e element
f[a],
p(a).
Proof. W e f i r s t admit 1 [a] = 1 . If p is a polynomial, we can find polynomials Q 1 , . , Qn with coefficients in A s u c h that
..
p(a)
- p(z)
(a 1- z 1) Q 1(z) +.
=
. .+ (an-z,)Qn(z).
Consider some u in S N ( a ;&), with N l a r g e enough. W e have ( a1, - z .1) d " u
=
. . , fid'lui-l n((ai-zi)d"ui)Ad"u.
d"ul A
A
. . . Ad"un,
and using (3.4.3)' we get (a,-z.)d"u 1
with v Using 1 [a]
=
yd"u 1
=
A
1
.. .
=
(-1)'dIlv A
n!
(p(a)-p(z))d"u A dz.
1 , we obtain
p(a) - p[a]
=
(&)n
C
Then the r e s u l t follows f r o m Ln
d"ui+l A
. . . Ad"un.
hd"ui-l
(ai- zi) Qi(z)d"u Adz =
and S t o k e s formula. We may calculate 1 [a] with and setting y. = 1
Ln
&. Choosing for
w e have
1
d(vQi dz)
i =1 , .
- ( a1, - z .1) u1'.
=
(al-Z1)U1 + y 1
=
( a l - z l ) u l + Yl((a2- z 2 h 2 + y2),
and by an e a s y induction
. ., n ,
some ui in SN(ai;
8,,)
SPECTRALTHEORY
36
1
=
(a-z,v)
..
with v1 = u , , v2 = y1ii2,. , vn SN(a ; 6,). Moreover n- 1 d"v = y 1 d"ul
.. . yn-,un.
y1
=
+ yl " ' Y n ,
d"u2 A
y:-2
A
Clearly v l , .
. . . A d"un
4
. . , vn belong to
.
T h u s the problem IS reduced to t h e calculation of ykd"u fi d z , where k is a non negative integer and u in S ( a ; with N l a r g e enough. For k = 0, we may choose N u in S 2 ( a ; i f w e can show that d'lu A d z h a s a limit when R t e n d s to infinity. Taking u = we have
Fa)
8,)
-z,!x
1
lirn R++m
Izl,(R
d"uAdz
6,,( =
=
lirn R++a
-dyz/\dz
IzKR
2 ( 1 + J z I )2
-2x1.
For k > O , w e u s e
T h i s is a consequence of S t o k e s ' formula and the following statement: Lemma 4.-
Let
u E S N (a; 6) and k b e a non negative i n t e g e r . I_f
W =
2
Proof. Obviously -
. . . A d"ui-l
uid"ul A
i=1
A d"ui+l A
d"w = n d"u. Write
d"u = ((a1- z1 )u 1 +. . .+ (an- zn)un + y) d"ul A
AS
(a,-z )d"u. 1
w e have
d"u
=
=
2
i=l
-d"y
1
1
=
(-l)i uid"y A W
-d"y
A
+ yd"u
-
jpi
d"ul A
. . . A d"u,,
.. .
A
d'lu,.
(a.-z.)d"u., J J J A
. . . A d"ui A . . . A d"un+
yd"u
.
Further (k+l)ykd"u = (n+k+l)yk+ld"u
-
( k + l ) yk d"yAW -nyk+'d"u,
and the statement follows a s d"(yk+'W) = ( k + l ) yk d"y A
doh+ nyk+'
d"u.
SPECTRALTHEORYMODULOAb-IDEAL
.-
37
o(8).Lf 7 a= . . . , x(a,,)), we
Proposition 1 1 L e t 6 be a weight function in A ( a ; A ) and f E bounded multiplicative linear f o r m on A , setting (a) = ( $ ( a l ) ,
have 8( %(a))2 0
;c
and
~ ( f C a 1 )=
Proof. Consider u
f(
X (a)).
in S N ( a ; g; A ) with N la r g e enough. Then ( ;C(u,),
belongs to SN($ ( a ) ; 8 ; C ) and 8 is s p e c t r a l for %(a) in C & ( ;C(a))>O and )I(f[a]) is equal to f [%(a)] =
(&In
nl
. .. , x(un))
. Th er ef o r e
d"( x ( u ) ) A d z .
l n
But ev er y neighbourhood of ;I(a) in C" is s p e c t r a l for $(a). Choose a polydisc D with cen t er a t )!(a) so that is compact in {8>0). Clearly f is the uniform limit of a sequence (p,) of polynomials on 6. Th e holomorphic functional calculus a t $ ( a ) being a bounded l i n ear mapplng f r o m into C , w e have
o(8,)
Using Proposition 10
, we get pn [;C (a)]
= pn( %(a))and f [$(a)]
= f(
X(a)).
3 . 5 . - S p ect r al theory modulo a b-ideal
We shall now examine what happens when a b-ideal I of the b-algebra of A is also considered. Definition 4 . - A non negative function if w e can find bounded mappings uo, u, v
0-f
8 -no
C n is s ai d to be sp ect r al for a modulo I , of Cn i s A and a bounded mapping
,. . . , un
C" G o I such that (a-z,
u>
+ v + Xu,
=
I.
Th e set of all s p e c t r a l functions modulo I for a is denoted by h ( a ; A/I). Th e pr o p er t i es of A ( a ; A) proved in Proposition 6 are extended without modifications to
A(a; A/I). Th er e also e x is t s a basis of &a; A/I) composed of Lipschitz functions I,, I where B is a bounded absolutely convex s e t i n A and B ' a bounded s e t in I , ( s ) is the distance in EB f r o m 1 to defined as follows: (f' 8, B
(a,- s l ) B +.
. .+ (an- s,)B
+ B' .
Hence A ( a ; A/I) h a s a b a s is of weight functions. When
8
is a weight function in A ( a ; A/I) and f E
o(S),we define
f [a]
in A/I.
S P E C T R A L THEORY
38
s; A/1)
T h e ldeds a r e similar t o those of Section 3 . 4 . W e denote by SN(a;
. .. , un,
..
the s e t of
v), where u ,,, , u ( r e s p . v) are continuously differentiable functions on C" taking t h e i r values i n A (r:sp. I ) , bounded along with t h e i r
all functions ( u , ,
derivatives of o r d e r 1 , and s u c h that
belongs t o
y
-Nzl( s;
S N ( a ; 8;A/I)
1
=
-
u>
(a-s,
-v
A ) . A s t r a i g h t forward extension of L e m m a s 1 and 2 shows that
is not void. However, i f
:I,
v are in S N ( a ; $ ; A/I) it is no longer
possible t o prove that f d"u A dz can be extended o v e r Cn so that i t is continuous and integrable. But i t is t r u e for f y d"u
A
dz
when N a P + 2 n + l , and w c s e t f[a]
(3.5.1)
=
&, f y d l ' u h d z .
(-12" ( n - t l ) !
(2~i)"
When I = 0 , because of Lemma 4 , t h i s definition is consistent with that of Section 3.4.
In the g e n e r a l case, the right hand s i d e of equality (3.5.1) is independent modulo I of ( u , v) i n SN(a;6 ; A/I) with N a P + L n + 1 . L e t u s c o n s i d e r t h e case n = 2 and keep the notations of Lemma 4 u i t h ( u , v) and ( u ' , v ' ) instead of u and u ' , W e define oc, a s previously and
El,
Then
e2
i
5, = v u i - v ' u l
c2 r\
=
v u2' - v ' u 2
=
yv' - v y '
u;-ul
=
(a2-z2)u+e,+<,
u;-u2
=
-(a,-~~)N+(~+t;~
vl- v
= -(al-zl)C
YV-y
=
)El
-(a 1- z 1
Only the c a s e where, for instance, cX= 0, explanation. W e have y ' d " u ' - yd"u
- (a2-z2)c2 + q - (a2-z2)C2 - Q .
e2 c1 c2 =0,
=
=
0,
9=O
-
=
r(y-(al-zl)~l)d"(ul+~l) yd"ul]~d"u2
=
[-(al-zl)fl d"u.,+yd"~,- ( a l - z l ) f l dI1$]Adl1u2
As
- ( a l - z ) d " u l A d"u2 1
r e q u i r e s some
=
d"y A d"u2
+ d"v A d"u2,
.
39
SPECTRALTHEORYMODlJLOAb-IDEAL
tends to zero in A when R tends to infinity. T h e r e f o r e
belongs to I
We are able now to give a generalization of the fact that the spectrum is never v o i d Proposition 12
.- rf 0 is spectral for a l ,.. . , a n- modulo I , then I = A .
Proof.- If OeA( a; we may choose y Then (3.5.1) yields As 1 [a] 1 , this implies modulo I , that is I A . A/I),
= 0.
1-0
=
I r a ] = 0 in A/I.
=
..
..
Consider now some b = (b , b ) such that b.1 5 a1. modulo I for i=l,. , n. If is s p ect r al for a modulo I , t h e r e e&t bounded mappings u,, u l , . , un ( r e s p . v) of C" into A ( r es p . I) such that
8
..
+ v + Su0
I.
=
Thi s can be written (b-z,
u > + v +
+ xu,
A s ai- bi belongs to I for e a c h i , obviously ( a - b , u
=
1.
>
is bounded in I and is also s p ect r al for b modulo I . Moreover SN(a; A/I) = SN(b;6 ; A/I) and f[a] = f[b] mod. I. W e have constructed a li n e a r mapping of $) into A/I. T h e r e is no natural boundedness on A/I; however the following property holds : if f remains in a bounded subset of and u , v are chosen in S ( a ; 8; A/1) for N l ar g e enough, the eleN ment fu[a] given by (3.5.1) is bounded in A and fu[a] fv[a] is bounded in I (where fv[a) is similarly defined).
s;
o(
o(8)
-
Notes S p ect r al theory of Banach a lg e b r a s is due to I . M . Gelfand ( I ) in the one dimensional case. The construction of the holomorphic functional calculus in the n-dimensional
case is due to G.E. Shilov ( I ) , R . A r e n s and A . P . Calderon ( I ) and L. Waelbroeck(T). The definition of s p e c t r a l sets and functions and the construction of the holomorphic functional calculus of b-algebras aregiven by L. Waelbroeck ( I ) ( 2 ) . An abridged version can be found in J.-P. Ferrier ( 2 ) . The exposition adopted h e r e is slightly different. O u r a i m h a s been to give a simple construction of f [a] when t h er e is no
40
SPECTRALTHEORY
ideal. It can b e shown that f 3 f [a] is an homomorphism, but the proof of L. Waelbroeck ( 1) by means of tensor products is r a t h e r complicated. W e have omitted such a property h e r e because we shall not use it. (")
Cn has been oriented s o that
(&)n
IT1
d z n dz is a positive measure.
CHAPTER IV
SPECTRALTHEOREMSANDHOLOMORPHIC CONVEXITY
At the beginning of the Chapter a f e w definitions and elementary propert i es in connection with plurisubharmonic functions and pseudoconvex domains
are briefly recalled. W e only insist on the equivalent conditions which lead 2 to the definition of pseudoconvexity. The L -estimates for the d" o p er at o r are used to study s p e c tr a l sets for z, , , , zn in Q6), when is a weight function. W e show that the s e t fl where 0 d o es not vanish is sp ect r al if and only if i t is pseudoconvex and apply t h i s r e s u l t to d i scu ss s p e c t r a l functions is f i r s t for z in The case when -log is plurisubharmonic in considered. To handle the g e n e r a l c a s e , we introduce a p r o c e s s of plurisubharmonic regularization of weight functions. A s an application of s p e c t r a l theorems, we obtain the existence on e v e r y pseudoconvex domain of an holomorphic function with polynomial growth which cannot be extended; we a l s o give a characterization of pseudoconvexity by means of bounded multiplicative linear forms.
..
F
o(s).
4.1.
- Preliminaries
Let
6
be a n open s e t in C". A function f defined in
n
with values in [-00 ,+cO[
is called plurisubharmonic i f it is upper semi-continuous and if for ev er y complex line
L , the r es t r i ct i o n of f to a n L is subharmonic. A non negative function f defined in fl is called log-plurisubharmonic if logf is plurisubharmonic in When f l , f2 are If f is analytic in then If I is log plurisubharmonic in , as it is upper log plurisubharmonic , then f , + f2 is also log plurisubharmonic in
-
a,
-
-
.
.
semi-continuous and a s the property is valid for functions whose logarithm is subharmonic. For instance, if
&, = (1 + Izl 2)-' '
is defined a s in Section 1.2, the function
SPECTRAL THEOHEMS
42
- log
F,
.-
Ilefinition 1 Ari open s e t conditions a r e fulfilled :
s
(1)
.
is plurisubharmonic in
H
n 1"Cn
is called pseudoconvex i f the following equivalent
(iii)
.
-logd(s,\hb) is plurisubharmonic in
T h e r e e x i s t s a plurisubharmonic function f vely compact in 0. for e v e r y real number c (11)
.
For e v e r y compact set K
1"a,t h e
hull
of K
< c)
is relati-
with r e s p e c t t o plurisub-
is compact.
harmonic functions in
r.
I e t K be a compact set in a n open s u b s e t a l l points s in
s u c h that {f
U J
of C n . W e denote by Kn the s e t of
n such that
-
.
It is the for every plurisubharmonic (or log plurisubharmonic) function f in Obviously is bounded hull of K with r e s p e c t to plurisubharmonic functions in is clobccausc> - log go tends to infinity a t infinity. It can b e e a s i l y proved that sed i n For e v e r y s 6 , t h e r e e x i s t s some plurisubharmonic function f defined m on such that f(s) c = s u f(<). A convolution argument by a f function which t;€ only depends on lz,\, , lznl , s h o u s that f is the limit of a d e c r e a s i n g sequence (f,) of plurisubharmonic functions which a r e continuous on a neighbourhood of K u i s ] . Hence, f o r n l a r g e enough we get
n.
>
n.
&
zn
E
.. .
and the property remains valid on a neighbourhood of s. W e f i r s t assume condition (i);then - l o g
- log 8,
( s ) = Max (-log d ( s
and condition (ii) follows a s
{8,<
is also plurisubharmonic in
,[a)-log ,
c) is compact in
If (ii) holds, for e v e r y compact s u b s e t K of
,-.
obviously Kn
c
n as
J0(s)),
for e v e r y r e a l number c .
a,setting
-
SUP (f(s)l s EK is contained in {f 6 c) ; t h e r e f o r e KO is relatively compact in =
I
and
w e obtain (iii). W e only have t o prove now that (iii) implies (i). It suffices to show t h a t , if D is t h e unit open d i s c in the complex plane, for e v e r y a , b E Cn s u c h that a+Bb C and e v e r y subharmonic function (4.1.1)
'4
n,
on a neighbourhood of a + b s u c h that
- l o g d ( a + t; b) ,< y(t;)
43
SPECTRALSETSFOR z
<
4
for e v e r y E d D , the same inequality is valid for e v e r y E D . Writing r e a l p a r t of some holomorphic function h , (4.1.1) becomes d ( a + < b ) > l e -h(C)
(4.1.2) Setting then
Fc(C)
=
a
I
4 b + e-h(c)c
+
a s the
,
a.
with c E C", I]cII< 1 , w e only have to show that Fc( a D ) C fi implies F c ( D ) C T h i s is t r u e for c = 0, and the s e t of all hero, 11 s u c h that Fhc(D)C for a given c , is open. The statement will be proved i f i t is also closed. L e t K b e the compact s e t with X6[0, I],
of a l l
i f F X c ( D ) c fl , a s f o Fk
and
FXc(e) E En
C E ~ D for ; e v e r y plurisubharmonic function f in
,
is subharmonic, w e have
; thus F k ( D ) C
n
implies F x C ( D ) c
En
,
Remark. We have considered in Definition 1 the distance d a s s o c i a t e d t o the hermitian norm in C"; the definition does not change Hhen d is the d i s t a n c e a s s o c i a t e d t o anot h e r norm, as the equivalence between conditions Proposition 1
.-
(iii) is s t i l l valid.
be a pseudoconvex open set i n C" aLd f b e a plurisubharmonic
function in
n ; the open s u b s e t w h e r e f
Proof. Let
K be a compact s e t in (f
is negative is pseudoconvex.
< 01 . T h e r e e x i s t s a h
f ,< c on K . T h e r e f o r e , f 6 c on the hull Kn , which is compact because functions in
-
(1) and (11) or
n
negative constant c s u c h that
of K with r e s p e c t t o plurisubharmonic is pseudoconvex. T h i s implies that
Kn C (f < 0). A s 2 , contains the hull of K with r e s p e c t to plurisubharmonic functions in {f < 03, the statement is proved.
4 . 2 . - S p e c t r a l s e t s for z @
We denote by
6
o(8).
a weight function on Cn and by
0 = {6 > 0)
2 does o(s).
t h e set w h e r e
not vanish. We want t o study the spectrum c ~ ( z ; o ( & ) )of z in t h e b-algebra A s u b s e t S of C" is s p e c t r a l for z in i f , for e v e r y s # S , we can find holomorphic functions u l ( s ) , , , un(s) on fl s u c h that
o(6)
..
(4.2.1)
and (4 - 2 . 2 )
(zl- s , ) u l ( s )
+.
SN 1ui(s)\ <
M,
..+ (zn- S n ) un(s) i =I , .
=
1
.. , n ,
where N is a positive integer and M a positive constant, both independent of s.
SPECTRALTHEOREMS
44
is denoted by ui(s;<), conditions (4.2.1) and
When the value of ui(s) a t ( 4 . 2 . 2 ) become (4.2.3)
(cTn-sn)u,(s;C)
u,(s;C;) +...+
(C,-S,)
=
1
and
SN(t) I U ~ ( S ; C ;,<) \ S C, c n .
(4.2.4)
for S ~
M , i=l ,
.. ., n
A s the left hand s i d e of (4.2.3) vanishes for
n.
<
= s , every spectral
a
set S for z
We are now asking for conditions e n s u r i n g that is s p e c t r a l for z. is pseudois s p e c t r a l for z in then It is f i r s t easily s e e n that, if convex. W e only have t o prove that KO is compact in for e v e r y compact s u b s e t K h If it is not compact, We a l r e a d y know that Kn is bounded and closed in of contains
o($),
7.
.
we can find a sequence
.
Cn in
tending to some boundary point
e x i s t s a positive number E such that Iui(s;c)l
<
S+E
< M’
=
C,, of a , T h e r e
on K and (4.2.4) yields M&-N
4
for c c K . Then ( u i ( s ; c )I M ’ and < $ - s , u ( s ; )> tends t o zero when p P i n c r e a s e s . W e obtain thus a contradiction with (4.2.37. W e s h a l l prove t h e c o n v e r s e contains and a ( z ; a K ) ) contains i t suffices statement; a s to consider the c a s e when 8 = Jn.
o(8,)
o(8)
cr(z;ab)),
a
Theorem 1 . - Let be an open s e t in Cn. T h e n only if i t is pseudoconvex.
Proof.
is spectral for z @
The n e c e s s a r y condition h a s a l r e a d y been s e e n . Assume now that
convex and s e t
o(b)ifd is pseud*
8 = sn . W e want to find holomorphic functions u l ( s ) :$I-) u l ( s ; c),. .
..., u ~ ( s ) : ~ F + u ~ ( son; ~n) s a t i s f y i n g ( 4 . 2 . 3 ) and (4.2.4). It is not difficult t o find differentiable functions u l ( s ) , . . . , u n ( s ) ; we may set for
instance (4.2.5) for i = 1 ,
. . . , n.
(4.2.6)
Obviously
- S,
( Z
and as
src,
=
n,C E ~we, have
,,h(s)>
JK)-
=
S(S)
1
s I< - sl
for s @! (4.2.7)
T h i s implies that
S‘ lohi(s) I ,<
1.
0 is always s p e c t r a l for
z in
“e,($).
SPECTRALSETSFOR z
45
But the coefficients ohi(s) a r e not holomorphic. In order to get holomorphic func2 tions, we have to modify them. W e s h a l l u s e the following r esu l t , i f I u \ denotes the sum of s q u a r e s of the absolute values of the coefficients of the differential f o r m u :
.- Let
b e a pseudoconvex open s e t in Cn , '9 a plurisubLemma 1 (L. Hormander) harmonic function in f), , and r a non negative integer. For ev er y differential f o r m v
n
of type (0, r ) 1" which is s q u a r e integrable with r esp ect to the measure e-ydh and s at i s f i es d"v = 0 i n the distributional s e n s e , t h e r e ex i st s a locally inteprable such that d"u = v @ differential form u of type (0, r+l)
(u12 e-'p$2 dX
<
[\vi2 e-(4dX.
If r , t a r e non negative in te g e r s , let
4.t denote the vector sp ace of all diffe-
re n t i al forms of type (0, r ) with coefficients in the ex t er i o r product At Cn which a r e s q u a r e integrable with r e s p e c t t o some measure SN d X Su ch a form h is a
.
skew-symetrical system (h ), where I r a n g e s o v e r the s e t of a l l multiindices I , it n. W e s e t (i.,, ., it) with 1 4 i.,,
.. . <
..
where I r an g es o v e r the set of multiindices (i W e define, for e v e r y s $0, an operator
'b
. , ., it) such that 1 < i l < . . . < i t6 n. t f r o m L r l t o Lr by
and set 'P h = 0 if h E L; , W e consider the double complex
Lt+l
I
r
sp
d"
'
Lt+ 1 r+l
d"
'
t Lr+l
where d" is the usual densily defined o p e r a to r which acts componentwise. I t is easi l y s
s
P P = 0, 'Pd" = d1ISP. W e have defined by (4.2.5) an element $ ( s ) = .(&.,(s), ,.hn(s)) of Lo1 It follows from (4.2.6) that Poh = 1 and from (4.2.7) that ev er y ,,hi(s) is s q u a r e N integrable with r e s p e c t to dX for N > / 2 n + 3 . Similarly &(s) belongs to t h e domain of dl' as verified that
. ..
8
.
SPECTRAL THEOREMS
'16
d"(.-Z)
z
1
I2
-s
1
- sl
=
-z. - -s.
-
-
1 1 - --(( z l - S l ) d z l + ...+ (Zn-Sn)dZn) + Iz-sl
dZi 2 ' Iz
- SI
W e f i r s t d e f i n e by i n c r e a s i n g induction a n element k h ( s ) of t h e domain of d " i n Lk+l s u c h that 'P h(s) = d" k - l h ( ~ ) k
P d"(k-,h)
T h e system (khl) is s k e w s y m e t r i c a l b e c a u s e k+ 1 t o the domain of d " i n Lk
.
=
0. M o r e o v e r , kh b e l o n g s
A s A"+'(C'') = 0 , be have ,lh(s) = 0. W e s e t n h ' ( s ) = 0 and define now, by i n c r e a s i n g Indiiction o n h , an element k h t ( s ) of such that d" k - l h ' ( ~ ) = kh(s) - " P k h ' ( s ) .
A s s u m e that r , h t ( 5 ) ,. . . ,
11
,)il
(s) are already defined. W c h a v e
' sN <
kh(s) - '1' ,,h'(s)l
dX
M
for some positive, i n t e g e r N arid some positive constant M . M o r e o v e r (1''
(,$s)
- 'P
kh'(s))
=
d 1 I k h ( s )- S P ( k + , h ( s ) - S P k + l h l ( s ) )
=
d " kh(s)
=
0.
Using 1,emma 1, w e c a n firid k - , h ' ( s )
-P '
in ' L :
k+,h(s)
s u c h that
and
W e finally s e t U(S)
ODviously u ( s ) s a t i s f i e s d"u(s) 1 belongs t o I,, , we have
- 'P
=
.h(s)
=
0 and
S
.h'(s).
Pu(s)
lu(s)l sN(s) d h
<
=
'P .h(s) = 1
and
, as u(s)
M,
for some positive i n t e g e r N and some p o s i t i v e c o n s t a n t M . T h e n P r o p o s i t i o n 2 of
47
S P E C T K A L FUNCTIONS FOR z
C h a p t e r I s h o w s that u l ( s ) , . . . , u (s) belong to O(F). I t is e a s i l y seen that a l l t h e e s t i m a t e s a r e independent of 5 . T h u s u l ( s ) , , . , u (s) a r e bounded independently of
.
O(8).
s in
i n Cn we c a n find a R e m a r k . Theorern 1 s h o w s that for e v e r y p s e u d o c o m e x open p o s i t i v e i n t e g e r h a n d a positive number M s u c h t h a t holornorphic functions u l ( s ) , . , un(s) c a n b e a s s o c i a t e d to e v e r \ point s 4 fl so that
..
(Z1-S
and
SE
1 u 1 (s)+. iu,is)i
. .+ (Zn-Sn)
,< M ,
i=i,
un(s)
=
1
. . . , n.
It IS important to note that t h e e s t i m a t e s u s e d i n thc, proof of T h e o r e m 1 a r e independent o f fl . T h e r e f o r e , me c a n find N , M s o that the. p r o p e r t y is valid for e v e r y
fl .
pseudoconvex open s e t
z . 3 . - S p e c t r a l functions for
z
o(8).
W e keep t h e notations of S e c t i o n 4 . 2 . A non negative function t r a l for z in
u,(s):
<
a$) for e v e r y if,
++ u o ( s ; c ) , . . . , un(s):
(4.1.1)
(Z1-S1)
'p
on C" is s p e c -
s € C", ue c a n find holomorphic functions HU~(S;<)
u , ( s ) +.. . * + (Z,l-S,l)
L
so that
in p
+
y(s, uo(s)
=
1
and
SN \ u l ( s ) \ <
(4.3.2)
M,
i=o,. . . , n,
~ h e r vN is a p o s i t i v e i n t e g e r and M a positibc c o n s t a n t , both independent of s.
Conditions ( 4 . 3 . 1 ) a n d ( 4 . 3 . 2 ) may also be Ltritten
( < l - s l ) ~ l ( s ;+. Q. .+ (Cn-Sn) ~ r l ( s ; c )+
(4.3.3)
and
XN(e)
(4.3.4)
for
seen, If
'4
M,
u,(s;C)
=
i=o, . . . , n
~ € 0 .
belongs t o
&;a&),taking l/CQ(C)
Then, using (4.3.4), w e get
or
<
I~~(S;C;)J
$4
=
s = C c n i n ( 4 . 3 . 3 ) we h a v e
u,(C;<).
1
SPECTRAI. THEOREMS
48
with
Ivy = SUP Ilb(s)\.
(4.3.6)
SE
n
EN I uo(s)\ ,< M , it is e a s i l y s e e n that
independentu,(s) is l o c ~ly bounded in ly of s ; t h e r e f o r e - log y is plurisubharmonic (and continuous). In p a r t i c u l a r N 2 I/M and e v e r y s p e c t r a l function for z is l a r g e r than some function equivalent tv As
8
8.
8 € A ( z ; a g ) ) , we get f r o m (4.7.5)
Moreover, i f
8
is equivalent to a function
Y
W e shall prove that such a condition
IS
Hence
is plurisubharmonic in s u c h that - log sufficient. W e c o n s i d e r a p a r t i c u l a r case
n.
8 b e a weight function on C" bounded by &, and s u c h that 8 is plurisubharmonic i n the s e t fi where $ d o e s not vanish; then 8 is s p e c t r a l
Theorem 2 (I. Cnop).- Let -log
for
o(F).
z
W e need the following Lemma 2.- Let
s be a non nepative function on Cn a n d n, denote the s u b s e t of
-
C"eC o f ( s , t )
wAh s E C n , t € C s u c h t h a t I t l < X ( s ) . T X :
s(s)
= d((s,o),[q),
where d is the distance with r e s p e c t t o the norm Moreover, assume that vex and that -log
8
(s,t ) e Is(+ It I 1_" Cn e C . that fi = {$ > 0 ) is pseudocon-
s is lower semi-continuous,
is plurisubharmonic in
; then
f),,is pseudoconvex.
Proof. By definition -
If
6 is lower semi-continuous,
a is open. Obviously, n1is the s u b s e t of
ax C
of
all ( s , t) s u c h that
-log
X(s)+log It1
6
4
0.
If -log is plurisubharmonic in , then (s, t) H harmonic in fl x C and Proposition 1 shows that
- log
Proof of Theorem 2 . W e keep the notations of Lemma 2 . A s
n
g ( s )+ log I t
nl is pseudoconvex.
fl
- log 8
is plurisub-
is plurisubharmonic
in and tends to infinity at the boundary, is pseudoconvex. Then t h e conditions of Lemma 2 a r e fulfilled and is pseudoconvex. Using Theorem 1 , we see that is s p e c t r a l for ( z , w ) in
a,
o(h)and, using Corollary 1 of Chapter 1
111, that a l s o
49
PLURISUBHARMONIC REGULARIZATION
o(
is s p e c t r a l for (z, w) in 8 ), where w denotes the second projection of C n e C . T h e r e f o r e , for e v e r y (s, t) ir? C'e C we can find holomorphic functions suchthat u i ( ( s , t ) ; ( < , T ) ) , i = O ,..., n+l in u i ( s , t ) : (<,T)I-.+
a,
.
( ~ , - s l ) u l ( ( s , t ) ;(C,T)) +. .+
(Cn-Sn) un((s,t); ( C , T ) ) +
+ and
for ( s , t )
E
Cne C and
. . ., n. B e s i d e s , a s
i=O,
x ( s ) = Min ( xo((s,O),d((s,O),
so,
R, n
{ o ] ) satistying u , ( s ; C ) f...+
(+,)
=
(5,-s,)
M
gN(s)\ u i ( s ; ~ ),< \ M , i=O,.
C E O ,and the proof
ui(s;c), i = O ,
u,Cs;l;)
and
for s e C n ,
F=
[q)) &+((s,O)).
W e have obtained holomorphic functions ui(s) : =
(c,
E u i ( s ; c ) = ui((s,O); 0)) for w is a weight function bounded by we have Min(&,,8).
Using then Lemma 2 , we get
n
(C,C))
1
=
< M,
(c,.) n,. We s e t 8
Un+l((s't):
8n,((s,t))uo((s,t);(C,
\ u i ( ( s , t ) ; (C,.C))l
g:l((c,Z))
('C-t)
+
.. ., n , in
8(4 u0(s;C)
=
1
.., n
of Theorem 2 is complete.
Remark. T h e positive integer N and t h e positive number M which a p p e a r in the above property are independent of the weight function
$ , when the
conditions ofTheorem 2
a r e fulfilled.
4.4.
- Plurisubharmonic regularization
W e need a definition. L e t
6
be a lower semi-continuous non negative function on
7
the w h e r e 6 does not vanish. W e denote by the lowest majorant of C" and denote ; s u c h a function exists: let $ s u c h that -log$ is plurisubharmonic in s u c h that f -log 8 Taking the set of all plurisubharmonic functions f i n
s2
<
i t is well-known that
'4
continuous, we have Cf
- log $
is upper s e m i -
be a lower semi-continuous non negative function on C" 8 d o e s not vanish is pseudoconvex. If -log 8 is pluriA , then also - l o g 8 5 8 satisfies W 2, then also 8 d o e s .
Proposition 2.- &t assume that the s e t subharmonic in
F
0 a n d , as $ = - log ? .
is plurisubharmonic in
< - log F . Then
.
0 where
SPECTRA1 , THEOREMS
50
I.emma 2. If -log 8 is plurisuhharmonic in 0 , then is pseudoconvex and the function ( s , t ) H l o g d ( ( s , t ) , is plurisubharu 5T monic T h e r e f o r e - log 8 _is plurisuhharmonic. F u r t h e r , if 6 %atisfies W 2 , then 6 x ; hence 8 . T h i s implies is a majorant of 8 But 6 is plurisuhharmonic in
Proof. T h i s is a n e a s y consequence of
.
.
-
[a,)
x<
h
s s a t i s f i e s W2. h
that
Proposition 3 . - I.et 2 he a weight func? does not vanlsh is pseudoconvex. Then 8
is a weight function and
o(8)
=
o(8).
on Cn arid assume that the s e t f ). w h e r e 8 0 on the complement of ,
, extended by
.
63
-.
We note that €8 f2r someApositive number & A s - l o g 6 is plurisuhharmonic, w e also have EO>,E8 A A s 8 s a t i s f i e s W 2 , i t is a weight function. C l e a r l y A and contains because 838. Conversely, let f be holomorphic in N satisfy 1 f I 8 6 M for some positive integer N and some positive number M . W e have 1/N (log I f 1 - log M) -logs
o(8)
O(8)
<
and, a s l o g ( f I is plurisuhharmonic
that is If
<
I$N
I/N ( l o g l f l M. Therefore
- log rvi)
o(8)
=
h
< -log$
,
o(s)and o($)have the
( 3 ( 6 ) . Moreover
same boundedness.
o(8)when the set where $
W e are now able t o c h a r a c t e r i z e the spectrum of z in
does not vanish is pseudoconvex.
8
6
'Theorem 3 . - Let he a weight function on C" and assume that the set where does not vanish is pseudoconvex. A non negative function Cf & Cn is s p e c t r a l for z
in o(6)if and only if
is l a r g e r than some function equivalent to
%; .
as).
Proof. F i r s t assume that 'p is s p e c t r a l €or z in We have a l r e a d y shown that > l / M SN where l o g y is plurisuhharmonic in fi . F u r t h e r , w e e a s i l y get 2 y 3 (1/M 2 l / M gN.
sN),
Conversely, let
get
s"
6,= Min ( 8, z0);then o(8)= o(8,) ~
Corollary 1
.- Let 8
he a weight function o n Cn
o(s)
and
denote the set where
not vanish; then 8 is s p e c t r a l for z & if and only if function y s u c h that -log is plurisuhharmonic in
that
and f r o m Theorem 2 we
A(~;o~J,)).
E
.
6
does
6 is equivalent to some
such The n e c e s s a r y condition h a s a l r e a d y been proved. If 6 is equivalent to A and t h e statement is a log is plurisuhharmonic, then 6 is equivalent t o
-
r
DOMAINS OF HO120MOIIPHY
51
consequence of Theorem 3 , A straightforward generalization of Theorem
j
is the following
Proposition 4 . - L,et A be a directed s e t of ueight functions. Assume that (3(0) complete and that { s , O ) is pseudoconvex for e a c h Sea. A non negative function (9 C" is s p e c t r a l for z 1_" t o some
3 with Xc A .
O(A)
if and only i f i t is l a r g e r than a function equivalent
he only note that cp is s p e c t r a l for z in
fc A
such that CQ is s p e c t r a l for
L
in
O(A) i f
o(s).
and only i f t h e r e e x i s t s some
4 . 5 . - Domains of holomorphy An important consequence of the r e s u l t s of the previous section is Theorem 4.- k t
0 be a
pseudoconvex open sct in C"; t h e r e e x i s t s a function f
of
o($n)which cannot be holomorphically continued beyond 0 . Proof. By virtue of Theorem 1 , the sct s p e c t r a l for z in o(80);t h e r e e x i s t s IS
a positive integer N s u c h that functions ul(s), so that ted to e v e r y point 5 4
( z l - s l ) u l ( s ) +.
(4.5.1)
. . . , un(s)
. .+ ( Z n - S n )
un(s)
Sn) can be a s s o c i a -
of
=
1
.
F i r s t assume by a b s u r d that t h e r e exist a connected open s e t 0 intersecting and a connected component w' of w n n such that, for e v e r y function g of Let w e can find some holomorphic function h on w which coincideswith g on a'. v l ( s ) , . . , vn(s) b e holomorphic functions on w a s s o c i a t e d to u,(s), ,u n ( s ) . By holomorphic continuation (4.5.1) should yield
.
.. .
( z l - s l ) v l ( s ) +. Choose now s
E
. .+ ( Z n - S n )
w n a n . A s ( z l - s , ) v l ( s ) +.
Vn(S)
=
. .+ ( z n - s n )
1*
vn(s) v a n i s h e s a t s , w e
obtain a contradiction. L e t (w r ) denote a denumerable b a s i s of connected open s e t s intersecting 2 0 and . . , denote the sequence of connected components of for e v e r y r let 0,n
1, n . C l e a r l up, y e a c h o(a,,) is a F r e c h e t a l g e b r a and e a c h fibrated product
is a FrPchet s p a c e . W e have s e e n that the f i r s t projection
5%
SPECTRAL THEOREMS
n h i c h is obviously continuous and i n j c c t i v c , is not onto. T h e Banach homomorphism
is theorem s h o w s that t h e union, when (r, p) v a r i e s , of t h e p r o j e c t i o n s of t h e E r tP go) which is contained i n t h e differvnt from L e t f b e a function of
(?(En).
projection of no E
r',p' S ~ ~ i p p o sbye a b s u r d that f c a n be holomorphically continued beyond a coiinectc,d opcv s e t w i n t e r s e c t i n g 261, a connected component 0'of holomorphic function g o n w s u c h that f
= g
on w'
. Clearly
: there exist
and a n
w i n t e r s e c t s w' and
.
[a' : as o
IS conncctcd, i t i n t e r s e c t s awl. C h o o s e then s e o n d d W e e a s l l y see that s can ; then s belong? to some W which is contained in 0 .A s U,no' is not void, it intersects some 0 and f = g on cd T h u s f b e l o n g s t o t h e p r o j e c t i o n of E r,P r,P' r,p'
fi of Cn is c a l l e d a domain of holomorphy i f w e c a n n o t find a ted opcri set o i n t e r s e c t i n g and a connected component w' of wnn s u c h t h e r e e x i s t s g E: o ( w ) with f = g o n w ' . If fi is a domain that for e v e r y f E of holomorphy, i t is ~ a s i l yp r o v e d that some f t c a n b e found so that n o w , An open domain
c orincc
o(n)
o(n)
w ' and g e x i s t mith t h e above p r o p e r t i e s ; i t is also p r o v e d that Lie obtain h e r e t h e well-known c o n v e r s e statement but T h e o r e m 4
fi is pseudoconvex. is m o r e p r e c i s e .
A g e n e r a l i z a t i o n to weight functions which a r e not n e c e s s a r i l y e q u a l t o some is given by
8*,
8,
Theorem 5 . - Let 8 b c a weight f u n c t i o n o n Cn bounded by and s u c h t h a t - l o g s 1s plurisubharmonic o n the o p e n s e t w & d o e s not v a n i s h . T h e r e e x i s t s a family (f,) of holomorphic functions i n such that N 1/g 5 SUP if,\ M/& ,
5
n
<
d
for some positive i n t e g e r N and some p o s i t i v e number M , which a r e independent of F r o m Theorem 2 , w e know that 5 is s p e c t r a l for z i n statement e a s i l y follows from (4.3.5) and (4.3.6) with 'Q = 6
-.
.
8.
O(g). T h e n t h e
1 . 6 . - Bounded multiphcative l i n e a r f o r m s We now g i v e a n e a s y application of t h e holomorphic functional c a l c u l u s . b e a weight function o n Cn s u c h that t h e o p e n set Thcorcm 6 . is pseudoconvex. For e v e r y bounded multiplicative l i n e a r f o r m )! 0" e x i s t s some s €0s u c h that
Proof. Upon
replacing
6
;C(f)
=
f(s) for e v e r y function f
h
by
6
we may a s s u m e that
- log 6
1"
= iSr0)
o($),there
as).
is p l u r i s u b h a r m o n i c i n
fl .
Then & is s p e c t r a l for z i n 0(g).T h e holomorphic functional c a l c u l u s a t z g i v e s into For e a c h s E , let a bounded l i n e a r mapping f H f[z] of
o(8)
o(g).
MULTIPLICATIVE LINEAR FORMS
53
')! b e the bounded multiplicative l i n e a r f o r m f w f(s) on 1 1 of Chapter 111 w e g e t , for e v e r y function f in
o($),
")! (fCz1)
=
o(&).From Proposition
fP)Xz , ) , . - . S )! (zn)), I
that is f[z] ( s ) = f(s). T h e r e f o r e f [z]
o(8).
=
f and the holomorphic functional c a l c u l u s is
the identity mapping of Now l e t b e a bounded multiplicative l i n e a r f o r m o n ( 3 ( S ) . Proposition 11 of Chapter 111 e n s u r e s that s = ( X(z,), , X(zn)) belongs to
fl
and that f ( f [z]) = f(s) for e v e r y function f in proof of the statement is complete.
. ..
o(s).Then
$f)
=
f(s) and the
Using Theorem 6 , we immediately obtain Proposition 5 the set
.- I&
$ > 0)
A b e a directed s e t of weivht functions s u c h that for e a c h
is equal t o some fixed pseudoconvex open s e t
n . For e v e r y bounded
multiplicative l i n e a r f o r m 7( 0" O(A), t h e r e e x i s t s some S E X ( f ) = f(s) for e v e r y function f 1"
an).
8~ A ,
n
s u c h that
As a consequence, we give a n o t h e r c h a r a c t e r i z a t i o n of pseudoconvex domains Proposition 6.- L e t
n
be a n open set in C";
e v e r y bounded multiplicative l i n e a r f o r m on
with s E . -
theno
is pseudoconvex if and only i f
o(n)is equal to some
'J
:f
t3
f(s)
Proof. If is pseudoconvex, we only have to apply Proposition 5 . Conversely, if is not pseudoconvex, t h e r e e x i s t s a compact s e t K in 0 s u c h that 2, contains a sequence (t ) which converges to some boundary point t of L e t 'u be a n ultraP f i l t e r on N which converges t o infinity. If f belongs t o a bounded s u b s e t of t h e r e e x i s t s some positive number M such that I f ( < ) \ ,C M for K. Therefore, P a s If1 is log-plurisubharmonic, by definition of Kn we have If(t ) ) g M. T h u s we
.
o(fl),
SE P
can s e t %(f) = l i m f(t ).
u
p
Moreover IX(f)I ,< M. Hence ;1 is a bounded multiplicative l i n e a r f o r m . As we see that X is not equal t o some with s E
sx
fi .
%(z)= t
Notes T h e equivalent definitions of pseudoconvex domains are well-known (see H . J. Bremermann (') ). T h e fact that a pseudoconvex open s e t is s p e c t r a l for z in
o(5,)
n
3 is a p a r t i c u l a r case, but with p a r a m e t e r , of a r e s u l t of L. Hormander ( )
..
mentioned as C o r o l l a r y 1 of Section 5 . 6 ; i t is the case when f l = zl-sl , , , f = z -s The proof is similar and the same double complex Lk is u s e d , I t depends n n n'
54
SPECTRALTHEOREMS
on the famous estimates of L. Hormander for the d " - o p e r a t o r (see L. Hormander ( I ) , (') and also J . B . Poly (')). T h e increasing and d e c r e a s i n g inductions are taken f r o m the original proof of Theorem 2 , or C o r o l l a r y 1 of Section 4 . 4 , by I . Cnop ('), (2), (3). H e r e the study of s p e c t r a l functions is deduced f r o m that of s p e c t r a l sets. T h e plurisubharmonic regularization and the o t h e r r e s u l t s of Section 4 .4 are d u e t o I . Cnop and the author ( l ) . T h e fact that pseudoconvex domains and domains of holomorphy are the same is a basic r e s u l t of complex a n a l y s i s in C" (see B.J. Bremermann ( I ) , F. Norguet ( I ) , K , Oka (2)). T h e existence for e v e r y boundary point of a pseudoconvex domain of an holomorphic function with polynomial growth which is s i n g u l a r a t t h i s point is due t o R . Narashiman ('), where t h i s domain is bounded. Theorem 4 for a bounded domain is given by N . Sibony ( I ) . Theorem 6 and Proposition 5 a r e d i r e c t applications of the holomorphic functional c a l c u l u s of b - a l g e b r a s , w h e r e a s Proposition 6 is well known (see R .C Gunning and H. R o s s i (I)).
-
.
CHAPTER V
DECOMPOSITION PROPERTYFOR ALGEBRAS OFHOLOMORPHIC FUNCTIONS
We introduce for polynormed algebras H of holomorphic functions o n a given domain fl , the following property : ev er y f E H vanishing a t s E can be written ( z l - s l ) u1 +. . .+ (zn-sn) un , so that u l , . , , un a r e bounded Such a decompoi n H when f v a r i e s i n a bounded subset of H and s in
a.
o(&)
.
sition property is proved for an algebra when 6 is a weight function and the open s e t 18>0] is pseudoconvex, and for a subalgebra of O(8) when 6, I a r e weight functions and the open s e t 42 7 0) is pseudo-
o(g')
o(A),
o(8')
convex. W e also r e p la c e 8(&), by inductive limits O(Al). F o r a g ene r a l H satisfying the decomposition p r o p er t y , we d escr i b e the r es t r i ct i o n t o
n of the spectrum of
z in
fi
and d i scu ss the s t r u c t u r e of H
when a condition of convexity of 0 with r esp ect to H is fulfilled. We apply these r e s u l t s to the study of b-ideals and finitely generated ideals of a l g e b r a s
06). 5.1.
- Preliminaries
o(0)
containing t h e b e an open s e t in Cn and H b e a subalgebra of Let polynomials, equipped with a n algebra boundedness. We s a y that H h as the decomposition property o v e r
fl if
the following condition is fulfilled :
F o r every function f @ H and e v e r y point s f = (zl-sl)
u, +.
so that u l , . .. , un are bounded i n H ,
& n.
& fl , with f(s) = 0,
. .+ ( Z n - S n )
when f
un
one can w r i t e
,
v a r i e s in a bounded su b set of H
and
s
56
DECOMPOSITION PROPERTY
I t is cquivalent t o a s k that for e v e r y bounded s e t R in H , t h e r e e x i s t s another belongs to bounded s c t B ' such that e v e r y function f i n B vanishing a t s E
) R' +. . .+ ( z n - s n ) B ' . 1 Note that the property depends on the boundedness of H. W e give a less rcstrictive condition which is actually equivalent t o the decomposition property. (Z - S
1
Proposition 1 . - Assume that for e v e r y f t H write f = ( 21- s 1) su 1
so that
8,N ( s ) 'u, , . . . ,
and
+. . .+
s
E
f l , wAh f(s) = 0 ,
(Zn-Sn)
sun
,
a r c bounded in H for some i n t e g e r N,
8 t ( s ) 'un
.
one can
whfn
f
v a r i e s in a bounded subset of H ,7nd s 1_" fl Then the same property holds for N = - I a n d H h a s the decomposition property o v e r
n.
}'roof. W e c o n s i d c r the coefficients -
'ul
and
-zi
=
sY
=
(1
'They obviously satisfy (2-s,
S,2(s) ,
S
+
(2,
u > +
i=1,..
z>) sY
., n,
2
SJS).
1.
=
Now let N b e a non negative integer such that f = (2-s,
where f and have
FF(s)' u l , . . . , f
and each
gr-'(s)
('U1
+Y '
vary
%> f + +
(z-s,
'u,)
,
u >
& r ( s ) 'un
= ( 2 - s , =
S
in a bounded s u b s e t of H . W e also
sY f
sY su
>,
v a r i e s in a bounded s u b s e t of H. By d e c r e a s i n g
induction on N , the statement is t h c r c f o r c proved.
o(Fd
Corollary 1 . - T h e algebra
-c".
of polynomials h a s t h e decomposition property
over
Proof. F i r s t let p b e a polynomial s u c h that p(0) = 0. One can h r i t e p
where q , , , .
=
z q
1 1
+...+ z n qn '
. , qn
.. .
are polynomials and q. only depends on z,, , z . . Moreover , . .'. , qn a r e bounded in when p is; i f r is a bound for the d e g r e e of p , t h e d e g r e e of e a c h qi is bounded by r 1 . Assume now that p(s) = 0 ; w e may c o n s i d e r t h e polynomial 'p = p(z+s) and w r i t e s u c h a decomposition is unique and q,
o(x0)
-
PHELIMINARIE S
57
a s above Thus p
=
If p is bounded in bounded in & A s ) 2r-1
x0);
(3(
( z l - s l ) s q l ( z - s ) +. ..+
o( so) and
. sq l ( z - s )
(Zn-Sn)
.
sqn(2-s)
r is a bound for the d e g r e e of p , then &o(s)r.Sp is F,(s)' . 'qi is bounded and so each
therefore e a c h 1s.
a / b z J is a bounded l i n e a r mapping of H
Corollary 2 . - Assume that each derivative
-
into H and let r be a non negative integer. In o r d e r that H h a s the decomposition
p r o p e r t y , i t suffices that for every f E H vanishing a t s along with i t s derivatives up to o r d e r r , one can write
when
bounded i n H ,
Proof. Suppose that -
as 'g(s)
=
=
0 and 'gl
-
f =
f ' . (z-s)+...+(-l)rf(r).
f(r'').
(2-5)'
derivatives up to o r d e r r. Moreover implies FT(s)Sg =
where ' u l , .
. . , 'un
+. . .+ ( z n - s n ) u n , so that u T , . . . , un a r e
n .
:1
f vanishes a t s and belongs to some fixed bounded subset of H .
Writing
sg
f = ( z l - s l ) u,
f v a r i e s i n a bounded subset of H a n d s
, clearly
( Z - S y ,
' g vanishes a t
s along with i t s
sF(s) ' g is bounded in H and the hypothesis
< z-s,
S
u
>
,
belong to some fixed bounded subset of H. Finally
f =
> +. . .+ (-l)r+l< z - s ,
(z.-s)r->
f(?
+
S,'(S)
su)
,
and the statement follows from Proposition 1
, the of a l l functions f in H vanishing a t s is equal to the b - ideal id1 (z - s;H) of G(8,;H) of all families generated by z l - s d , . , zn- sn. Moreover the ideal H)) : if a family (Sf)sfn such that f(s)= 0 is equal to the b - i d e a l i d l ( z - s ; , t h e r e e x i s t s a positive i n t e g e r N such that ('f) belongs to a bounded subset of 8,N (s)'f l i e s in some fixed bounded s e t in H and N If H is a complete subalgebra with decomposition property, for e v e r y s E
ideal
'3
. .
3
8,(S)Sf =
where ' u l , .
,
. , 'u
,
also lie in some fixed bounded s e t in H. Then as Sf =
clearly f'
c(so;
< z-s,$,N(s)Su)
is bounded in id1 (z - s ;
implies the decomposition property.
,
c(so; H)). Conversely 3
=
id1 ( z - s ;
c(L,;H))
DECOMPOSITION PROPERTY
58
5 . 2 . - Decomposition property for (3 (g)
I
T h e fact that (3(8,) h a s the decomposition property is nothing but a p a r t i c u l a r c a s e of the following r e s u l t : Theorem 1
6
s
.- Let
does not vanish is pseudoconvex. T h e n
a.
Proof. Upon
where
b e a weight function on Cn and assume that the open set
replacing
8
o($)
h a s the decomposition property o v e r
f
by the weight function
defined in Section 4 . 4 , we may
assume that - log 6 is plurisubharmonic i n 0 , W e introduce another v a r i a b l e ; now s = ( s , , , , sn) denotes the first projection of C" x C" and z = ( z l , , , , zn) t h e 1 second one. Using Theorem 2 of C h a p t e r IV and identifying with t h e b - a l g e b r a of all functions in 6 o ) which only depend on z, we know that is s p e c t r a l
.
for z in
o( s o(8 e 5 ), Applying then the holomorphic functional c a l c u l u s , we define for o(5)a n element f[z]
e v e r y function f in of
x
.
o(6)
0
and
K
of
o(6 s ). Let
(d,
B
b e the bounded multiplicative l i n e a r f o r m f u f
c ) b e a n element
( u , 1; ).
From
,c)
Proposition 11 of Chapter 111, we have % (f [z]) = f ( 'j (z)), that is f [z]( 6 = f(c). Hence f[z] = f ( z ) . Similarly is s p e c t r a l for s in 6 e ) and f[s] is defined
o( 6
8
= s modulo the b - i d e a l I
and s a t i s f i e s f(s] = f(s). But z
=
idl(z-s;
f[z] - f [s] is bounded in I when f v a r i e s i n a bounded s u b s e t of can find g l , , , gn in a bounded s u b s e t of @ ) s u c h that
..
o(6
f(z) - f(s)
=
(zl-sl)gl
+. . .+
o(8 e 8 )) and
O(8). We t h e r e f o r e
(zn-sn)gn.
Returning t o our previous notations, l e t s denote an a r b i t r a r y point in
en,let
'gi = gi(s, z) and assume that f(s)= 0. Then (5.2.1)
f
..
=
o(g);
,
Obviously ' g l , . , gn are in they are not bounded when s v a r i e s but t h e r e N exists a positive integer N s u c h that e a c h ( s )'gi is bounded in Using again the fact that ficients 'u,, (5.2.2)
'ul,.
S
F
o(8).
8
SN
and thereby a r e s p e c t r a l for z in (3($), we obtain coefbounded independently of s in 2) s u c h that
. .,'u
n' (z-s,
Su
>+
o(
SN(S)SU, = 1 ,
Multiplying (5.2.2) by f and using (5.2. I ) , we get f =
Su
f
>
+
< z-s,
%> ,
su, gN(s)
and the decomposition property is now proved. A s a n immediate consequence of Theorem 1, we obtain the following more g e n e r a l statement:
59
DECOMPOSITION PROPERTY Proposition 2 . - L e t A be a d i r e c te d s e t of weight functions such that for each l c
. Then o(0)has
the set { 8 > 0 } is equal to some fixed pseudoconvex open set the decomposition property o v e r fl
.
For instance, when
is pseudoconvex, the algebra
tion property; f o r e v e r y convex increasing mapping (3(h ) a l s o has. o, 'p
,
o(n)h as the decomposi-
of [O,
00
C
into
[o, 00 [ ,
5.3. - Decomposition property for subalgebras Let 8 , $ equivalent to Theorem 2 . -
be weight functions on C" such that
8 If
and f l n l
F1
is l a r g e r than some function
n1denote the s e t where st does not vanish. O(sl),equipped
is pseudoconvex, the algebra
o(z),h a s the decomposition property o v e r n1.
n e s s induced by
with the bounded-
W e need the following Lemma 1
.-
f be a n holomorphic function defined on a pseudoconvex open s e t
w
g&
with l/lf(q)I
sf
1 m f f vanishes a t = 0 if 7 4 ~ E. n If I holomorphic functions u, , , u & W such that
. ..
S G (3
, t h e r e ex i st
1- s 1) u 1 +...+( z - s n ) u
f = (2
g&
1 u ~ 1 S f \ 1 " <M,, s
i=T,
e No, M, only depend on n.
Proof.
W e note that
Obviously If
ITf <
8,
is the weight function
'pf
1 and If
= Min
..., n
vf,
where 'Qf is defined by
( l h l , 2 ,).
18, 4 1 ; thus
f belongs to
a(6,)
and satisfies in
a n estimate independent of f .
0 (6,)
As -log '9 is plurisubharmonic in 0 , Proposition 2 of Chapter IV s h o w s that also - l o g is. If f vanishes a t S E o , applying Theorem 1 i n we obtain holomorphic functions u, , ,un on W such that the conclusions of Lemma I hold. It
8
...
o(8,)
is clear that No and M, only depend on n , because the constants appearing i n the property
o(s,)
lfE A ( z ; o(s,))and the holomorphic o(8, .a 8 f ) , only depend on n.
3
functional calculus
60
DECOMPOSITION PROPERTY
o(
nl
Proof of Theorem 2. Let f be a function of &I) vanishing a t s E and assumc that f belongs to a bounded subset B of (3(g) : th er e ex i st a positive integer N and a positive number M such that
If15N 4
(5.3.1)
Using Lemma 1 , we can write in and satisfy
SF
(5.3.2)
From (5.3.1) , w e get &SN is bounded by
S"'lfl,(
sN f o r
E
u>
i=1,
where u l , .
S N N o lull
. ., u n
a r e holomorphic
..., n ,
sf
E 7 0 small enough and we may assume that N 'Qr 2 and a l s o 2
go and s a t i s f i e s W L . Then
.., un
< M,
lgN
~2 .
-No
O($)
a r e bounded in when f v a r i e s in B. Reasoning si m i l ar , from for some N ' , M I depending on f , we deduce that
M'
for some
< z-s,
l u l / < M,,
1/1 f I >/ E
Thus ( 5 . 3 . 2 ) implies
and u l , .
f =
M.
> 0 also depending on f . Hence u 7 , . .., un belong to
I
a(8 ').
More generally, w e may consider directed s e t s , A of weight functions such that each E A' is l a r g e r than a function equivalent to some 86 A . Then
-
Theorem 3. Assume that e a c h > O } , y& 6 E A , is pseudoconvex and contains . Then a fixed open set A'), equipped with the boundedness induced by O(n), has the decomposition property o v e r 0, 9 fi 3 { 8 > 0 ) for each S E A
n
For instance, let
o(
.
A be a directed s e t of weight functions such that for each $e A ,
the set { s > O ] is equal to some fixed pseudoconvex open s e t 0 ; l et K , K ' be compact s u b s et s of such that K ' is a neighbourhood of K . Every function f E
o(A)
a
vanishing at s € 0 can be written
so that gl ,
. . ., gn belong to o(a) and are uniformly bounded on .
K when f is
o(
uniformly bounded on K ' To prove such a property, we may consider on A ) the str u ct u r e induced by 0( $2 ) and apply Theorem 3 , If f is uniformly bounded on K I , i t is bounded in g,,
. ..
o(6kl); we therefore can find
o($
.
gl,. , , gn in
o ( A ) so that they
). But sk is uniformly bounded f r o m below on K and , gn are uniformly bounded on K .
a r e bounded i n
l
SPECTRAL FUNCTIONS FOR z 5 . 4 . - S p ect r al functions for z
61
o(
W e consider an open s e t fl in C n and a subalgebra H of 0)containing the polynomials, equipped with a n algebra boundedness We assume that the s t r u c t u r e of H is finer than the s tr u c t u r e induced by O ( n ) and that H h as the decomposition
.
(1
property o v er
. Such an algebra will be called a subalgebra
with decomposition
property of 0) A A s H may not be complete, we introduce the b - algebra H . For ev er y point s in 0 the multiplicative l in e a r form 'J : f I-+ f(s) is a bounded l i n ear f o r m on H ; 0 t he r ef o r e ST can b e continued as a bounded multiplicative l i n ear f o r m on H . A s
O(
s x (f)
tive. Let
S
=
(g) implies f
A
=
g w h c n i n H , the n a tu r a l homomorphism H -+ H is injec-
s e n and B be a n absolutely convex bounded subset of H such that
we denote by
'3
EB;
1E
the ideal of H composed of a l l functions f su ch that f(s)= 0 and in the pseudonormed v e c to r sp ace EB, f r o m 1 to the l i n ear
(a); the s e t
Proposition 3 . - Let H be a subalgebra with decomposition property of 0 of all functions 8, is a b a s is of the r e s tr i c t io n to of the spectrum of z
Proof. We f i r s t show that e v e r y function 8, *
is the r est r i ct i o n to
function for z in H . Choose E e l 0, 13 and let
SE
n
2.
of a s p e c t r a l
0 ; by definition of
S
I J
sB(s),t h er e
e xi s t s some &u0 in B such that 1
belongs to
'3
- ( S B ( S ) f t ) :uo
. C l e a r ly
=
zf is bounded in H as S,(s)
the decomposition property, we w r it e Zf
..
,"f
=
>
< z - s , u:
is bounded by 111I] B . Using
,
where & u l , . , &un belong to a n absolutely convex bounded su b set B' of H which and E > O . Thus does not depend on s 6 S
S
+
S
SB(S)ZU0
= 1
-
6
:uo.
Applying now Proposition 5 of Chapter I1 and using a n argument si m i l ar to that of Proposition 4 of Chapter 111, we c a n find an absolutely convex bounded subset
BIl
A
of H
such that 1 belongs t o (zl-sl)B"
+. . .+ ( z n - s n ) B " + gB(s)B" ,
for e v e r y s 6 fi T h e r e f o r e 8, is the r e s tr i c t io n to of some function in A(z; G). A fi , 'un , bounded in H Conversely, l et 5~ A ( z ; H). We can find coefficients 'u,, independently of s and such that
a
.
.. .
(z-s,
su
>
+
&s)su,
=
1.
DECOMPOSITION PROPERTY
62
L e t B ' be an absolutely convex bounded s u b s e t of H such that e a c h 'ui A
unit ball of E B l : for e v e r y
E > 0, t h e r e e x i s t
(I ;ui - su.IIfi < 1 EBI
su,, Ful,.
. . , :un
belongs t o the
i n B s u c h that
,
E
i = 0 , . . . , ri. Choose another absolutely convex bounded s u b s e t B of H s u c h that B contains B 1 u z, 8 ' u u z n B ' ; e a c h multiplication by z1 continuously maps
for
.. .
n
P.
A
E B l into E B , and therefore E B l into EB. In E B , for s fixed in
But < z - s, :,I r e s t r i c t i o n of
>
5
to
belongsto ' 3 and Zu,, to B. Hence is l a r g e r than some function
6,.
fl
We now give another characterization of
.n
s,
n
, we
get
8 ( s ) 3 gB(s) and the
.
Proposition 4 L& B b e an absolutely convex bounded s e t of complex functions ; then defined on
Proof.
.
Writing f = f(s) + g , we Consider a function f in B and a point s in define some g in By definition of ( s ) , t h e r e e x i s t s a l i n e a r form /A on EB B vanishing on and such that llpII< 1 , p(1) = ZB(s).Then = f(s)p(1) and
"3 .
3 '
Taking the supremum on f
p(f)
c B , we obtain
To prove the c o n v e r s e inequality, w e set 1/s
=
sup If fEB
I.
Obviously B is contained in t h e s e t of all complex functions f on If
18< 1 .
such that T h e r e f o r e the Minkowski functional of B is l a r g e r than the pseudonorm
and
Hence, taking
<
=
s , we get S,(S)
2 Z(s).
CONVEXITY
63
From Propositions 1 and 4 , we immediately obtain
o(a);
Theorem 4.- k t H be a subalgebra with decomposition property of a basis * of the r es t r i ct i o n t o fz of the spectrum of z & H is composed of all functions I / s u p If1 , w h e r e B is a bounded s e t in H. f€B
- Convexity
5.5.
with r e s p e c t to a lg e b r a s of holomorphic functions
Keeping the notations of the preceding section, we d i scu ss convexity of
0 with
re s p ect to the subalgebra H. Theorem 5.- Let H be a subalgebra with decomposition property of
o(n);the
following p r o p er t i e s a r e equivalent : (i)
T h er e ex i s t s a bounded subset B
(ii)
0-f
H s uch that for ev er y boundary point s
s u p If I is not bounded in any component of near s. f€B A is pseudoconvex and is a subalgebra of H with a finer s t r u c t u r e .
- fi , the function of
o(Sa)
(iii) C!, is s p ec tr a l for z
&H . A
Proof. First assume that (i) holds. Then for e v e r y connected open s e t w intersecting and ev er y component w of a n ( ] , the projection O(fi)ll,,(3(ce,) can-
an
-PO((-))
not be onto; i f i t was, by virtue of the Banach homomorphism theorem, it would be a n
O(n),
isomorphism of F r e c h e t algebras. As B is bounded in i t would also be bounded in o ( W ) and s u p ( f I would be bounded in the neighbourhood of ev er y point s Hence is a domain of holomorphy and thereby pseudoconof w n awl in of vex.
.fEB
8
Theorem 4 shows that t h e r e exists some weight function in A (z; 6) such that vanishes on the boundary of fl ExtenBeing continuous, 1 / sup 1 f I on " functions by 0 on the complement of fi , we obtain a bounded l i n ear ding
.
2<
.
o(rfi) o(g).
mapping of into Besides, the holomorphic functional calculus a t z A .+ H W e have already noted that for ev er y yields a bounded l i n e a r mapping A s E fi , the bounded multiplicative l in e a r f o r m ")c : f w f(s) can be continued to H ;
O($) .
moreover, from Proposition 11 of Chapter 111, we have for ev er y f in
" I (f[zI 1
= f("% (2)) =
o( n ,
o(s)
f(s).
A
) -+ H is injective and multiplicative; we Hence the bounded l in e a r mapping may identify with a subalgebra of H with a f i n er boundedness and condition (ii) follows.
o(A)
Obviously (ii) implies (iii)because Theorem 1 of Chapter IV shows that 0 is ,spect r al for z in Finally (iii) implies that so is spectral for z i n H. Using Theorem 4 , we can find a bounded subset B of H s u ch that 4 su p If and fsB
o(&).
l/gn
DECOMPOSITION PROPERTY
6.1 condition (i) holds. We s e t a definition.
o(n);
Definition 1 .- Let H be a subalgebra with decomposition property of equivalent conditions of Theorem 5 are fulfilled, we say that 0 is convex wlth
H - convex.
respect to H ,
n is H-convex,
i t is pseudoconvex. Conversely a pseudoconvex 0 is convex 1s a \ t i t t i respect to 0 or more generally ev er y O(A) where = {8 > 0 ) for each 8 E directed s et of weight functions such that I e t H be a subalgebra of O ( n ) such that 0 is H-convex. Asanimmediate /-. consequence of Proposition 3 and Theorem 5 (iii), a b asi s of the spectrum of z in H is the s et A of a l l meight functions Min ( Zn, as is pseudoconvex and, log If I is plurisubharmonic in fi , each - l o g Min ( , is -log $ = sup a l s o plurisubharmonic i n fi Every absolutely convex bounded s e t B in H such that 1 E EB is bounded in O(Min ( Ln, $,)); therefore the identity is a bounded l i n ear mapping of H into 0 (A). A Further the holomorphic functional calculus yields a bounded l i n ear mapping +H If
o(s,),
(a)
n
.
gB);
n
.
such that H
o(b)+,
4
& s),
o(0)
A
H is the identity mapping.
o(n)
such that H a subalgebra of Proposition 5 . - Let fi be a n open s e t in C n 0 2 H is equal to f), & H - convex. Every bounded multiplicative l in ear mapping
some "x
: f ef(s)
have s =
sE
.
A
can be continued to H. In view of Proposition 1 I of Chapter 111, w e
W e note that A s f[z]
with
) ( ( Z ) E ~
and
X(f[z])
=
f ( x(z)) = f(s) for ev er y function f i n
O(A).
= f when f belongs to H, the statement is proved.
When H
=
A
H , w e get
Theorem 6. - k t H be a complete subalgebra with decomposition property of O(n). c o n si st s of all weight functions @ - H-convex, then H = where
gfl
M i n (&,
O(A),
c,).
fl
denote a pseudoconvex open set in C"; complete subalgebras with decomposition property H of O(n) such that is H convex are exactly al g eb r as O(A), where A is a directed set of weight functions and (1 is the set where each Let
-
8€ a
do es not vanish.
5.6.
- Ideals of
holornorphic functions
W e consider in this section a n open set
decomposition property H of
a
(3(n)such that
in Cn and a complete subalgebra with is H-convex.
65
IDEALS O F HOLOMORPHIC FUNCTIONS Theorem 7 . -
b - ideal of H ; i f t h e r e e x i s t s a bounded family (h,)
I
-
1" 1
such
that
1" n , we
>/
SUP lh,l have I
=
1
H.
Proof, W e can e a s i l y find a bounded family
.
1 in 1 such t h a t 'h(s) (Sh)sca Multiplying 'h by 1/ 'h(s), we obtain a bounded family; we t h e r e f o r e may assume that 'h(s) = 1 . Then 'h - 1 vanishes a t s and is bounded i n H when s r a n g e s o v e r
12
>
. Using the decomposition p r o p e r t y , one can w r i t e
1s
( z - s , u'
(5.6.1)
>
+
'h
-
1
< z-s,'u >
=
, that
1,
=
.. .
'h
.
, 'un a r e bounded i n H independently of s E is s p e c t r a l where 'ul, As for z i n H, we can find coefficients S u l , . , 'u bounded in H independently of sE such that
..
[n
(5.6.2)
1.
=
From (5.6.1) and ( 5 . 6 , 2 ) , i t is immediately s e e n that 0 is s p e c t r a l for z in H modulo I . By virtue of Proposition 12 of Chapter 111, t h i s implies 1 = H.
8
C o r o l l a r y 1 (L. HSrmander).- Let be a weight function on Cn s u c h that, U I equivalence, - l o g $ is plurisubharmonic on 0 = > 0) and l e t f l , , f m belon_g to The b - i d e a l generated by f l , . , f m is e q u a l to only if t h e r e e x i s t some positive i n t e g e r N and some E > O such that
- o(s).
..
I f l / +...+ I f m J
Proof. If
idl(f
1 = f g I 1
+...+
>,E8
f
o(&)
o(6)
N
.
,,... , f m ; O($)) = O(Z), we can find
that
. ..
gl,.
. . , gm
in
O($)s u c h
g
m m'
..
,m. E>O s u c h that E x N Ig. I ,C 1 for j=1,. J T h e r e f o r e If, I +. .+ I f m l >/ f '. Conversely, assume that s u c h a property holds. Using Theorem 5 of Chapter IV, we c a n find a bounded family (g ) in (3(8) s u c h P that T h e r e e x i s t a positive integer N and
.
Setting now idl(fl,.
o(
= (j,
e)
Swig I >/ P P
and h,
..,f m ; o(6))and
i d l ( f l,...,f,,,;
SLP I
m/E8
N
.
f . g , obviously (h,) is a bounded family in I f,l>J P1. From Theorem 6 we then get
=
06)) = 06).
66
DECOMPOSITION PROPERTY Corollary 1 can be applied to algebras a s
0(Sn)
or
o(
fl is a [O, w[ into
) when
pseudoconvex open s e t in C" and rfl a convex increasing mapping of 10, m [ such that 8 1s a weight function. fly
'0
-
'Q be a non bounded Corollary 2. I,ct fl be a pseudoconvex open set in C" convex increasing mapping of (0,03[ [o, coC . Let also f , , , , f m b e e -
..
+
mentsof O(A ) . T h e n i d l ( f l , ..., f m ; O ( h )) QQ Q(f only i f t h er e exist positive numbers C , c, € s u c h that
Proof. W e
have already seen in Section 1.5, that
Xc
is pseudoconvex, -log&
As
I,
exp (- 'p (-log c
=
O(A )).
o(A
=
cp
is plurisubharmonic in
)
Q?
=
o[(Kc)),
)=
where
-
and as cf
is convex, also
~ ( - l ~ g c $= ~-log ) h, is. From Proposition 2 of Chapter IV, we know that - l o g x c is plurisubharmonic in 12 T h e condition of Corollary 2 is obviously n ecessar y ;
.
\a~lF
.
converselyit implies that I f l ) +. .+ I f m for some positive i n t eg er N and some € 7 0 , c > 0. Using Corollary 2 , w e see that i d l ( f ,, , , , f m ; (3( contains
1 . Hence id1 ( f l
, . . . ,f m ;
0(A
n*'4
x,))
.
)) also contains 1 and the proof is complete.
A generalization of Theorem 7 is the following
Let I & I s u ch that
Theorem 8.(h,)
in fl , then
b - i d e a l of H s;P
k
and
g E H ; if t h er e e x i s t s a bounded family
lh,l>lgl
g E I for some positive integer k .
Proof. W e can find a bounded family s E fl
. Using the decomposition g-
..
S
(Sh)sen in I such that 'h(s) property, we write
h =
'U
>
=
g(s) f o r ev er y
,
.
where 'ul,. , n are bounded in H independently of s E W e introduce the algebra Hl = HLX] of polynomials with coefficients in H , equipped with the natural boundedness described i n Section 2 . 5 , and the b - ideal
I1 = 1 i(1 - gX) H l . Clearly +
S
hX + ( I - g X ) = 1 ,
-
and 0 is s p ect r al for z in H1 modulo I1. T h e r e f or e I, = H, and 1 g X is invertik ble modulo I[X] T h i s implies that g is nilpotent modulo I and that g E I for some positive integer k.
.
67
IDEALS OF HOLOMORPHIC FUNCTIONS Reasoning l i k e for Corollaries 1 and 2 , w e deduce from Theorem 8 Corollary 3 (J.J. Kelleher, B. A . Taylor, I . Cnop) Cn such that, up to equivalence, -log g a r e functions of 0 (g) such that If11
+...+If,
6
.- Let 8
be a weight function on
is plurisubharmonic. Assume that f ,
I
E.SNIgl
, . . . , fm,
,
for some positive integer N and some € > 0. Then there exists a positive integer k k belongs to the b-ideal of generated by f , , . , f m .
o(s)
such that g
Corollary 4 .
- J& fi
be a pseudoconvex open set in C"
convex increasing mapping of a r e functions of
. .
O(A
If,/
P
[O, w [ into ) such that
+...+If,
1
for some positive numbers C , c ,
&
gk belongs to the b - ideal of
>
o(A
E
[O,
and
'p be a non bounded
co [ . A s s u m e that
f1,
. ..,fm,
g
exp(-Cy(-logc&))/gI
. Then there exists a positive inteqer ) generated by f 1 , . . . , f m .
k such that
n y c p
Notes The decomposition property for algebras of holomorphic functions, equipped with
o($)
boundednesses, has been introduced by the author ( 2 ) and used to study algebras in the one dimensional case. Similar ideas a r e developed here in the n -dimensional
-
case. Theorems 1 and 2 have been proved by the author ( 3 ) , (4)by means of the double
complex Lrt of Section 4.2. Diagram chasing was f i r s t used in this context by L .
Harmander ( 3 ) to prove Corollary 1 of Section 5.6 and by J. J. Kelleher and B.A. Taylor ( 1 ) to prove Corollary 3 . The method adopted here, based on the holomorphic
-
functional calculus modulo a b ideal is due to L. Waelbroeck. Corollary 4 of Section 5 . 6 has also been obtained by J. J. Kelleher and 8 .A . Taylor in the case where
fl
=
C"; our proof, based on plurisubharmonic regularization, is simpler.
CHAPTER VI
AI'PROXIMATION THEOREMS
We define the hull of a compact subset K of a given domain
with res-
pect to a given algebra I3 of holomorphic functions on . In the c a s e when (2 is C" and H is the algebra of polynomials, we c h a r a c t e r i z e the polynomially convex hull of K by means of s p e c t r a l theory of Banach a l g e b r a s and give a short proof of the so-called Oka-Weil Theorem. Using the r e s u l t s of C h a p t e r s IV and V , w e d i s c u s s the case when H is the algebra of a l l holomorphic functions on a pscudoconvex domain
o r , more generally, a
a
-
subalgebra H with decomposition property s u c h that is H convex. T h e r e s u l t s are applied to approximate holomorphic functions on the neighbourhood of a compact s e t , to study Runge domains, Rungc p a i r s and g e n e r a l i z e Runge property. W e f u r t h e r extend the theory to approximation with growth, W e consider a weight function $ and d i s c u s s density in of a subalgebra
o($)
H with decomposition p r o p e r t y ; in p a r t i c u l a r , when - l o g s is plurisubharis equivalent to the following convemonic in is>0) , density of H in
o(s)
xity hypothesis : up to equivalence, I/$
is the supremum of moduli of functions
of H . Another equivalent condition is given when H is equal t o some O ( b l ) , in p a r t i c u l a r when H is the a l g e b r a of polynomials, and examples of s u c h a situation a r e considered, in connection with a l g e b r a s (3 ( A+ for instance.
O(e- IZip),
o(8w ),
6 , l . - Approximation on compact s e t s In this s e c t i o n , w e consider a n algebra H of holomorphic functions on a given open , If K is a compact s e t in fl , we define the H-convex hull of K as the set set K H of a l l points s in s u c h that
n
APPROXIMATION ON COMPACT S E T S
69
for ev er y function f in H ; we s a y that K is H-convex if KH = K . When fl = Cn A / and H is the algebra of polynomials, we write KH = K p ; it is called thepolynomially A convex hull of K ; if K = K p we s a y that K ispolynomially convex. When H is the fi A algebra of a l l holomorphic functions on , we write K, instead of K an). A Obviously KH is always closed i n 0 If a compact set K is H - convex, a funA
O(n)
n
.
damental system of neighbourhoods of K consistsof s u b s e t s of
0
defined by inequali-
ties lfllC l , . - , l f m l < l ,
where f , ,
. . ., f m
belong to H
For suitable al g e b r a s H, we may c h a r a c te r iz e the H-convex hull by means of spect r al theory. As a d i r e c t application of the theory of Banach al g eb r as, we r ecal l the following statement : Proposition 1
on K
.-
k t K be a compact s e t i n C"; A
of polynomials, w
Proof. F i r s t -
m Kp
=
if PK denotes the uniform cl o su r e
s p ( z ; PK).
K p is contained in s p ( z ; FK).If s belongs to K p , consider the multiA plicative l i n ear form K : p I-+p ( s ) ; i t follows f r o m the definition of K p that id is continuous on the algebra of polynomials equipped with the uniform norm on K; thus
r
A
A
can be continued t o
PK and as
o g Z I ) , ..., % ( Z n ) ) s p ( z ; PK). K p contains s p ( z ; PK). If s =
c lear l y s belongs to W e now show that
A
e xi s t s some multiplicative l in e a r form ;C on Then P(S) = But the norm of
I
P(;C(Zl)'.
7
s belongs to s p ( z ; PK), t h e r e
nK su ch that
. ., ;I(z,))
=
s
=
( ) I ( z , ) , . . .,
(2,)).
1(PI.
is bounded by 1 and
Th er ef o r e s belongs t o Kp.
-
Corollary 1 (Oka-Weil). k t K be a polynomially convex compact s e t in Cn holomorphic function on a neighbourhood of K ; then f is a uniform limit on K polynomials.
Proof. A s
of
f
a"
K = s p (z; FK),the holomorphic functional calculus en ab l es u s to define
APPROXIMATION THEOREMS
70
f [z] in the Banach algebra
PK. From Proposition
1 1 of Chapter III, we get
f [z] ( s ) = f(s) for e v e r y point s in K . T h e r e f o r e f [z] is the r est r i ct i o n of f to K
and the statement is proved.
We want to extend Proposition 1 to more g e n e r al al g eb r as H. W e have used the polynomials only when writing p( X(z)) = %(p). Suppose now that H is the algebra O(n)and l et denote the uniform closure of on a compact subset K of CZ If 1 is a multiplicative l in e a r form on , i t is continuous on
(3(n),
.
- o(n) o(n),
when equipped with the topology of uniform convergence on K . Then
O(n),
;c
is a bounded
n is pseudoconvex, f r o m Proposition 6
multiplicative l i n ea r form on O(n).When of Chapter I V , we get x(z) E and )C(f)
=
f ( j ( z ) ) . Th er ef o r e
- K be a compact subset of a pseudoconvex open set
Proposition 2 . - & I
; then
Fu r t h er Corollary 2 . - Let K (3(n)-convex compact subset of a pseudoconvex Open set 0 and f a n holomorphic function on a neighbourhood of K ; then f is a uniform l i m i t
0" K of polynomials. More generally, if
RK
denotes the uniform c lo s u r e on K of functions of H , we have
Proposition 3.- &t be a n open set in Cn and H a subalgebra with decomposition property of such that g H-convex; for ev er y compact subset K 0,f , A we have KH = s p ( z ; BK).
O(n)
It is an e a s y consequence of Proposition 5 of Chapter V . From Proposition 3 we deduce Corollary 3 . - We keep the assumptions of Proposition 3 . &t
-
K H convex compact subset of 0 and f an holomorphic function on a neighbourhood of K ; then f is a uniform limit on K of polynomials.
6.2.
- Runge domains and generalizations
We f i r s t establish the following
n
Proposition 4 . - &t be an open set i h C" ; the following p r o p er t i es a r e equivalent : * (i) K p n n is compact in , for e v e r y compact su b set K offl (ii)
n A
(iii) K p
.
is pseudoconvex and the polynomials are dense in
is contained i n
, for e v e r y compact
subset K
O(0).
.
RUNGE DOMAINS
71
Assume that (i) holds. It is clear that the hull 8, of e v e r y compact su b set K /.. with r es p ect to plurisubharmonic functions in 0 is contained in K p n ; t he r ef o r e fi is pseudoconvex. In view of Proposition 1 , the spectrum of z in P, is A K p and the holomorphic functional calculus of Banach al g eb r as gives a bounded l i n ear
Proof. of
,-.
mapping O(Kp)+ PK.A s K P n a is closed in K extending functions by z e r o c\ PA we get a bounded l i n e a r mapping o ( K P n n ) .+ (Kp). Composing then with the A na tu r al mapping n ), we obtain a bounded l i n ear mapping A
A
o(Q)*O(Kp
o(0)
0
n
,
-+ PK , which coincideswith the r e s tr i c t io n to K because f[z] ( s ) = f(s) for e ver y s E K , by virtue of Proposition 1 1 of Chapter 111. Thus ev er y function of is uniformly approximable on K by polynomials and condition (ii) is proved.
o(n)
be a multiplicative l i n ear f o r m on the Banach of functions to K maps into PK and ;C defines a bounded multiplicative l in e a r f o r m on O(n). A s fl is pseudoconvex, Proposition 6 of Chapter IV shows that X (z) belongs to Thus contains the spectrum of z A in PK which is also Kp. F u r t h e r (ii) implies (iii). L e t
algebra
(3(a) n
PK. The r e s tr i c t io n
.
A s (iii)obviously implies (i), the proof of Proposition 4 is complete,
fl s at i s f i es the equivalent conditions of Proposition 4 , we s a y that e open set. Another characterization of Runge open setsis given by
If
m
Proposition 5
where
.- Let 0 a n open set i n C n ;
fl
is a
-
is Runge if and only i f it is H convex,
H is the algebra of polynomials equipped with the s t r u c t u r e induced by
O(a).
Proof. W e note that H h a s the decomposition property over fi in view of Theorem 3 A of Chapter V . If hz is H-convex, it is s p e c t r a l for z in H = R; for every compact set K i n , we have a natural morphism 3 PK and the spectrum of z in P, is contained in Then condition (iii) is fulfilled. Conversely, condition (ii) of Proposition 4 implies condition (ii)of Theorem 5 of Chapter V; i f is Runge, i t is therefore H convex,
n
.
a
-
Reasoning similar and using Proposition 2 instead of Proposition 1, we easi l y obtain Proposition 6.
- LA
fl , n
be open s e t s in C" s u ch that
is pseudoconvex and
; the following p r o p e r t ie s are equivalent
contains A
(ii)
is compact i n fl , for e v er y compact su b set K K o(sl,)nfi is pseudoconvex and O(nl) is dense in O(a).
(iii)
K O (sll)
(i)
r*
is contained in
, for e v e r y
compact subset K
of
0-f
n.
a.
When the equivalent conditions of Proposition 6 are fulfilled, we s a y that is a Runge p ai r . We also have
(a, 0)
APPROXIMATlON THEOREMS
72 Proposition 7 .- Let contains 0 ; then
n, nl (n,R
be open sets in C" such that
is a Runge p a i r i f and only i f
o(fl') equipped with the s t r u c t u r e induced by
€I
I)
nl
is pseudoconvex and H - convex, where
0g
0 (n).
W e finally give a general statement including both Propositions 4 and 5 which is a consequence of Proposition 3.
n
Proposition 8.- I,et 0 , nl be open s e t s in C" such that Q' contains and H be a subalgebra with decomposition property of su ch that fl & H - convex; the
o(a!)
following p r o p er t i e s a r e equivalent : A
K H n n is compact in
(i)
n , for e v e r y compact su b set K of fi . O(n).
is pseudoconvex and H is dense in
(ii)
KH is contained in
(iii)
(iv) induced by
& H,-convex,
o(a).
fl,for e v e r y compact subset
K
of a .
where H I is the algebra H equipped with t h e s t r u c t u r e
6 . 3 . - Basic approximation theorem
s.
We study now approximation with r e s p e c t to some weight function The conveof Section 6 . 1 will be replaced by the condition xity hypothesis I / x K = s u p 1 f,l
I/S
= sup
I ~ J up , to equivalence on
F .
.- LA $ be a weight function on Cn fk a n open set such that 0 ) . We consider a subalgebra H of o(s) o(f).,) containing the 3 W polynomials and assume that H h a s the decomposition property o v e r n , when Theorem 1
fl
=
equipped with the s t r u c tu r e induced by equivalent : (i)
as). Then the following conditions are
T h e r e ex i s ts a bounded family (f,)
( i ' ) T h e r e e x i s t s a famlly (f,)
l/S
&H
such that
8 >, (1/
su p
I fd) over 0.
i&H s u c h that, up to equivalence =
sup ( f J
over 0 . (ii)
( 111) "'
O(6)and,up to equivalence, 8 is s p e c t r a l for z & R. =
is plurisubharmonic in
-logs
n.
'
Proof. As aF)is complete, we recall that sa r i l y induced by
=
h
'i is not neces'i into
H; the s t r u c t u r e of
o($),but the identity mapping is a morphism of
o(8).
Using Propositions 3 and 4 of Chapter V , we first prove that conditions (i), (1') and (iii)are equivalent. If (i)holds, let B be a n absolutely convex bounded s e t in H
APPROXIMATION WITH GROWTH
73
s&
.
s u c h that 1 E B and f d e B for e v e r y d W e have 5, a n d , a s H h a s the decomposition property o v e r (3 , c l e a r l y is the r e s t r i c t i o n to w of some s p e c t r a l function for z in R. But (i) implies condition (i)of Theorem 5 in C h a p t e r V and 0 is
6
5
H-convex. T h e r e f o r e w is s p e c t r a l for z in R and also is. F u r t h e r , i f (iii) holds, as H h a s the decomposition property o v e r 0 , t h e r e e x i s t s some absolutely convex bounded s e t B in H s u c h that 1 6 B and 1/s
,< s u p
fEB
If1
5 3 sB
on
n . Then
a
T h e r e also e x i s t a positive i n t e g e r N and a positive number M s u c h that e a c h f E B N If M . Hence
satisfies
1s <
and condition ( i ' ) follows. W e finally o b s e r v e that ( i t ) implies trivially (i).
O($),
We now show that conditions(ii) and (iii) a r e equivalent. Assume (ii); as R = we only have to prove that is s p e c t r a l for z in ) ; as up to equivalence, - log is plurisubharmonic in 0 , t h i s is nothing but C o r o l l a r y 1 of C h a p t e r IV.
F
o(8
8
Conversely assume (iii). Using the n a t u r a l morphism
spectral for z in
O(s). T h u s ,
R +. o(s),
up t o equivalence, -log$
6
we see that
is
.
is plurisubharmonic in 0
Moreover the holomorphic functional calculus gives a bounded l i n e a r mapping j R , which coincideswith the identity mapping; t h e r e f o r e =
o($)
<3(8).
6 . 4 . - Approximation with growth W e c o n s i d e r the case when the subalgebra H of (u&)is equal to some O(A'), is larger 8' 6
where A t is a d i r e c t e d set of weight functions s u c h that e a c h than some function equivalent t o 8
.
18' > 0) is pseudoconvex for e a c h (3(A') is d e n s e i n o( 6 ) :
Theorem 2 . - Assume that the open set T h e following conditions imply that (i) Up to equivalence, I/: each f, belongs t o O(A').
gt
.
is t h e supremum on { 6 > 0 ) of a family ( [ f m l ) , w h e r e
(ii) Up to equivalence, l/g is t h e supremum on { 8 > 0 ] of a family (&), e a c h na L a log-plurisubharmonic function in some g), with sdc
where
c( A. Proof. By virtue of Theorem 3 of C h a p t e r V , the a l g e b r a ant),equipped with t h e = {8> 0 1 . T h e s t r u c t u r e induced by O(&), h a s the decomposition p r o p e r t y o v e r (J
first p a r t of Theorem 2 is t h e r e f o r e obvious : condition (i) is nothing but a reformulation of condition ( i t )of Theorem 1. Moreover (i) e a s i l y implies (ii). W e only have to
74
APPROXIMATlON THEOREMS
prove the converse statement. Assuming that (ii) holds, we define a weight function Yci
by
y,
Min ( ( I / % ) - ,
=
6,) ,
where I/R= has been extended by 0 on the complement oi
a,
n,= { 1$,>0)
I
As
a,
is pseudoconvex and - log ( 1 /na) plurisubharmonic i n , also -log (1/ ndl and -lo g y, are. Using then Theorem 5 of Chapter I V , for each d. we can find a family
(fm,P)P
in
o(-f') such that IVY, <
If,,pl
s;P
I
. We
where N is a positive integer and M a positive c o nst an t , both independent of may assume that P I/S s u p ITw 1/LS
a
for some positive integer P and some E > 0, and that t is such that weight function bounded by A s jTa I/y.(, obviously
is a
<
a
so.
As
I/%
> ESP,
wealso have
<
<
Y,>Esp
Therefore the proof w i l l be complete if e a c h f
c(&),
El8
and
@,P
belongs to
o(& ) ; as
x, is in
it is easily shown that >&gP,where & , P depend on a and that ya is l a r g e r than some function equivalent to s, ; then o(yb) is contained in
O(S,).
In the particular case'when
o(A')
is the algebra
o(8,)
of polynomials, we
obtain Corollary 4.- The following conditions imply that the polynomials are d en se in
o(g):
(i) Up to equivalence, 1/8 is the supremum of a family of moduli of polynomials.
(ii) Up t o equivalence, 1/8 is the supremum of a family of log-plurisubharmonic
functions with polynomial growth on C".
We immediately list a few examples.
.
1) Let a be a positive number and = e-"la Maybe weight function, but it is homothetic to some one, Obviously
is not exactly a
APPROXIMATION WITH GROWTH
75
and each
is log-plurisubharmonic on C n and h a s polynomial growth at infinity. Th er ef o r e the polynomials a r e dense in the algebra of e n t ir e functions of o r d e r d
.
2) Consider in C" the polyedron IP11< I , where pl
, ... , p,
defined by inequalities
G,
**.,
IP,l
9
-
a r e polynomials: if d denotes the distance to the complement of the W is supposed to b e compact
unit d i s c in the complex plane, w e s e t , when = Min(dop
dop,).
Obviously $ is homothetic to some weight function. I t is easily s e e n that W is exactly the s e t 0) Moreover
Is>
.
- c)-',
and each (z with IC 1 , is a uniform limit on the unit d i s c of polynomials, Then we can find a family ( q S ) of polynomials such that
on the unit d i s c. T h e r e f o r e
on
o , and the polynomials are dense in
o($).
-
We can obtain through this method a new proof of the Oka Weil Theorem. Assume that f is holomorphic on a n open neighbourhood fl of some polynomially convex compact subset K of C n . W e can find a polynomial polyedron G) su ch that K c w c c sd If 8 is the function defined above, as f is bounded on w' , obviously f belongs to (3(g). Then f is the limit in of polynomials; but convergence implies uniform convergence on K. in Instead of a polynomial polyedron, we may consider a polyedron 0 defined by
.
o($)
O(X)
inequalities
IflI <
I,
..., I f m ] <
1
,
,. . .,f m are holomorphic functions on a pseudoconvex open set . W e thus obtain a new proof of Corollary 2 .
where f l ning 0
of
b e a pseudoconvex open set in Cn and 3) L et [O, +co[ into [O, +a[ such that
'Q
contai-
a convex increasing mapping
APPROXIMATION THEOREMS
76
is equivalent to some weight function. For e a c h positive i n t eg er p , l et be equal (pP on to [O, p] and to some affine function tangent to Q a t p on [p, + g o [ It is easily s een that
.
S,,yp belongs to
c(ga)
using Theorem obtain that
(ii),
=
and that log ($
-'
exp (cQp(-log
) is plurisubharrnonic in
n, 'PP
we see that
o($*)is dense in o( 0
convex increasing mapping of
(0,
+a[ i n t o
algebra defmed in Section 1.5. W e recall that
1, Fi x c>O; choose
cQp
Q
=
exp (- y ( - l o g c
varies' we
'p
o(
'4
)).
a s in example 3 ) and s e t
xp =
exp ( ( ~ ~ ( - 1 o g c X Q ) ) .
c(gQ)
W e see that R p belongs to and that plurisubharrnonic in Moreover
n.
1/xc
Then
,S
'f
be a non bounded A ) be the Q 'p is the s e t of all
in Cn . Let [O, +a[ and
h
a . As When
o(&$ is a l s o dense i n o(n).
4) W e again consider a pseudoconvex open s e t
and
8fi))
=
log
xP
=
yp(-log
$a- log c)
IS
s u p TT P P
o(&)is dense in o(x",). As the property is valid for ev er y o($*)is dense i n 6(An, g).
c > 0, the
algebra
Notes The r e s u l t s exposed in Section 6 . 1 , except Proposition 3 and Co r o l l ar y 3 , and in Section 6 . 2 , except Proposition 8, are classical and most of them are due to K Oka L)
( I ) ; applying h e r e spectral theory, we obtain v e r y sh o r t proofs. Approximation theorems with growth are developments of i d e a s of the author ( 3 ) , (4).
CHAPTER VII
FILTRATIONS
We have s t u d ie d in Chapters IV to VI the b - algebra
o(S),when
is
a weight function. H e r e we are interested in the polynormed algebra O(S); instead of taking isomorphisms, we c o n s id er the s t r u c t u r e given by the sequence (NO($))of Banach s p a c e s , In this context, we apply a p r e c i s e s p ect r al theorem to prove p r o p e r ti e s of plurisubharmonic functions on a pseudoconvex domain : for instance, if a plurisubharmonic
7.1
.- F i l t r at ed b - a l g e b r a s Let _E = ( E , (NE)NEz) be a polynormed vector sp ace defined by a covering
indexed by Z , such that e a c h NE is a Banach s p ace and the identity NE j N+,E ha s norm ,C 1 ; we s a y that is a filtrated b s p a ce .If B is a bounded set in &, the
-
smallest NE 2 such that B is bounded in NE is called the filtration of B and denoted The filtration of an element x s E by Q(B); if 8 is not bounded, we s e t Y ( B ) = +03
.
is the filtration of (XI ; i t is denoted by v ( x ) . Consider a l i n e a r mapping u of a filtrated b s p a c e E into another F. W e s a y that u h as a finite filtration i f t h e r e exists some k E Z such that u is continuous from NE into N+kFfor ev e r y N E Z o r , equivalently, if
-
Y(u(B))
4
Y ( B )+ k
,
FILTRATIONS
78
for every bounded s e t 8 in E . The smallest k such that the property holds is the
filtration V(U) of u . Let F be a l i n e a r subspace of a filtrated b - s p a c e E W e naturally equip
.
F with NF of
a s t r u ct u r e of filtrated b - s p a c e , by considering for each N G Z , the cl o su r e F n NE in N E , The identity mappings F +- F and F+ E have non positive filtrations. We s a y that F is dense with filtration in E if F is dense in E and if the identity mapping E 3 F h a s a finite filtration, T h i s means that ev er y element of NE is the limit in N+kE of elements of F , where k is independent of N. A filtrated b - s p a c e A , fitted out with a s t r u c t u r e of algebra is said to be a filtrated b - algebra i f
N A * F>A
N+pA
?
for al l N , P in 2, and the multiplication NA x F ,A + N+PA h as norm ,C 1 . For instance are filtrated b - a l g e b r a s , when
c(s),o(s),"e,(s), Gcr($)
is a weight function
such that
1,
-
7 . 2 , S p ect r al theorem with filtration
-
.. .
s
, an be Let A b e a commutative filtrated b a lg e b r a with unit element and a l , on Cn is sp ect r al for a l , ,% elements of A. W e s a y that a non negative function
s
.. .
with filtration if t h e r e e x i s t s some positive integer k su ch that for ev er y positive int e g er N one can find bounded mappings u,, u l , . ,
(a 1- z 1) u 1 +...+ ( a n - z n ) u n +
., un
of Cn into N+kA satisfying
S N u,
=
1
and
.-
Theorem 1 I& $ be a weight function bounded by 1 0" Cn and does not vanish; t . n is s p e c t r a l for z with filtration in -log is plurisubharmonic in
Proof, Clearly -
s
$
the set where
o($)if and only if
.
is s p e c t r a l for z with filtration in
o(2)if t h e r e e x i s t s some
positive integer k such that for e v e r y positive integer N and ev er y s E C" one can find functions
(7.2.1)
uo(s) :
H
u,(s;~),
.. ., un(s) :
( c , - s l ) U 1 ( s ; t ) +. ..+ ( C n - s n ) u n ( s ; c ) +
and
(7.2.2) with
t+
$N+k(c) ui(s;c)
=
u,(s;&
s
of
o(i)satisfying
N (5) uo(s;c) = 1
0(cN), i = o , . . ., n ,
SPECTRALTHEOREM
c p
lim N++m
79
= 1 .
F i r s t assume that such a property holds. Taking s =
in ( 7 . 2 . 1 ) we have
~N(c)uo(<:c)1, =
and using ( 7 . 2 . 2 ) w e obtain
l/sN(d) ,<
(7.2.3)
IUO(s;t)\
S € i l
,<
cN/xN+k(C).
These inequalities c a n also be written -log
5 < yN <
+
-(l+R k ) l0gX
log CN
with
Let then N tend to infinity; as log C N and converges to - l o g
n , a l s o - log S
$
k tend to zero, (fN uniformly
n . As yN is plurisubharmonic
on e v e r y compact subset of
is.
We suppose now that -log L > O . For a given s in
is plurisubharmonic in
, we consider the
N and
'f2 ,.
=
=
W e choose a that
ewpartition of unit
,
("9 , "
D
(7.2.4)
and fix a positive i n t eg er
covering of Cn by
{ S 7(1+E)s(s))
sn 1XC(1+2E) s
and
n
(s)}
a
subordinated to su ch a covering so
= O(l/E$(s))
for i = 1, 2 and ev e r y derivation D of o r d e r 1. For instance let 2 sing mapping of R into R such that p(x) = 0 for x 6 7 and Using Proposition 4 of Chapter I , we find a weight function
x
'9
,< ( I + & / 4 )
s
p bea p (x) = 1
st
ern
increafor x >/ 1 . such that
is uniformly bounded for e v e r y derivative D of o r d e r 1 . Then
and that D
S
ern
< F'
in
- 'vl
s a ti s f y the r e q u ir e d p r o p e r ti es, for & small enough. is plurisubharmonic in fl , the open set 0 = { 6 > 6 ( s ) ) is pseudoconvex. In view of Theorem 1 of Chapter I V , we know that 0 is s p e c t r a l for z i n
and
= 1
As -log
2
o(Su); as s d o es not belong to
S
U1,...,
on
w s u c h that
(J ,
, we can find holomorphic
functions
FILTRATIONS
80
1 and
,
< z - s , SU>
=
s,". I 'ui\
,<
M,,
i=l,
..., n ,
where No, M, are universal constants. Moreover when < E
-
&l;, and thereby if
ES
m
2
&)
d(C)-J(C9) <
Hence the distance f r o m sufficiently small so that
2(Z(C)-2(C9
s ,< so
JN0 I 'Ui I ,< M,(
1 =
c<'a,
1 =
<
by z e r o on n n
that is
'y,,
tli
=
>
+
.
su(c)>/
'f + '(fa,
=
su s'Q1>
2 - s , s"
if
sv.
&/2)-No
and using 1
2-s,
4
0
< 2 I<-
w is at least , we therefore obtain
l o the complement of
E
I
we have
E&Q)
1
€kC)< &L;9+
Extending 'Ui
n, ,
+
s(f2
&/2 E/2
. (c).If
Cll
E is
F(c)and
we have
1
XN(s)%,,
i=l,
...,n
and S v,
=
P ( s ) s'Q2.
W e easily see that
(7.2.5) and, as
'vi
FN0
,< (1+2 L )
(7.2.6)
i=t,
= O(
..., n
2 ( s ) on the support of 'v, , that p sv, 0 ( ( 1 + 2 c ) N ) . =
.
Coefficients 'v,, 'vl,. . , 'vn a r e not holomorphic in ,< ( 1 + 2 E ) s ( s ) on the support of d" 'vi = Ui d" ST (7.2.7)
gN+No+2
dit
svi
o ( c ~8 N+l(s)) ,~ ,
=
with 'N,C
=
-No- 1
(1+2€
, but differentiable; as , using (7.2.4) we obtain
.
i = l , . , , n,
IN.
Using Proposition 2 of Chapter I , it is easy to transform uniform estimates (7.2.5) to (7.2.7) into L2-estimates. F i r s t considering i = 1 , . , n, for a sufficiently l a r g e positive integer kl , only depending on n, we obtain
..
(1
Id" 'viI2
S2(N+k')dX
=
O(CN,c
$ N+ 1( s ) ) .
SPECTRALTHEOREM
81
By v i r t u e of H o r m a n d e r ' s d" - L,emma, t h e r e e x i s t s a locally integrable function 'wj on such that d" 'wi = d" 'v.1 and
n
Therefore
(//Swi\2 5
We now s e t
Obviously
u. = 1
S
ul,
5
v.
1
;2(N+kf+2)d), )1/2
- sw 1.
. . . , 'un
for i = l ,
..., n
g(s)
6
Iz-sl
and
s(s)-N(l- ( z - s ,
5u0 =
su
> ).
and also 'u,
are holomorphic in
<
su, = svo t J ( s ) - N As
O f C N , E gN+'(s)).
='
2 - s , sw
>
is. Moreover
.
is bounded, we have
~(S)-~(S I
12(N+k'+3) dX
'w>I
From (7.2.6) we deduce a s i m i l a r estimate for
S
=
vo; then 'u,
O(CN,E
*
also satisfies s u c h a n
estimate. Using again Proposition 2 of Chapter I , we can transform this l a s t L
2
-
estimate into a uniform one. For a sufficiently l a r g e positive i n t e g e r k , only depending on n, we obtain
Reasoning similar for i = 1 ,
. . . , n,
we also have
Choosing now E = 1/N, we have
.
When g ( s ) = 0, we can c o n s i d e r and the statement is proved when s belongs to the coefficients u l ( s ) , , un(s) given by Theorem 1 of Chapter I V ; they s a t i s f y
...
(zl-s,) ul(s) and
+. . .+
gk ui
(Zn-Sn)
= 0(1)
,
un(s)
+
x
N
..., n ,
i=l,
for some positive i n t e g e r k; then we also have $N+k ui
and the proof is complete.
=
O(I)
,
i = l , ..., n
,
(s) =
I
a1 7 . '3
FILTRATIONS
.- Application
t o plurisubharmonic functions
A consequence of t h e methods developed in S e c t i o n 7.2 is T h e o r e m 2.bounded and
Let $
b e a L i p s c h i t z non negative function o v e r C n s u c h t h a t t h e set w h e r e d o e s not v a n i s h ; t h e following conditions are
s
equivalent : (i)
.
is plurisubharmonic i n
-log
for e v e r y positive i n t e g e r N , t h e r e e x i s t s a family (f,) (ii) functions i n so that
n
'/SN 6 w h e r e k is independent of N
Ci
9
I r~ 1 ,<
cN/gN+k
of holomorphic
9
and lim C y N N-?+rx,
1
=
(iii) t h e r e e x i s t a family (n ) of p o s i t i v e i n t e g e r s a n d a family (gg) of holomorP p h i c functions i n so that -logs
= sup
B
l/ng
log
.
\"a\
Proof. Assume that (i) holds. For a sufficiently s m a l l p o s i t i v e number 1 , c l e a r l y h& is a weight function s u c h that 1 . A s -log is p l u r i s u b h a r m o n i c i n fl Theorem 1 shows t h a t A 6 is s p e c t r a l for z with f i l t r a t i o n in As). T h e r e f o r e (7.2.7) c a n be w r i t t e n
1s
<
1/ w h e r e e a c h u,(s)
AN SN
<
SUP l ~ o ( s ) I
S€L!
i s holomorphic i n
<
cN/
a n d lim
N++m
o(
,
SN+k
CAiN
=
1. A s
and
condition (ii) is p r o v e d . Supposing now (ii), l e t (f
W e immediately have
N . a )u
b e a family of holornorphic functions i n
0
such
PLURISUBHARMONIC FUNCTIONS
p= (N, a ) and
and condition (iii) follows with
gB
=
83
l/CN f N , a
, as
Finally (iii) obviously implies (i), Proposition 1
.- Let a be a pseudoconvex open s e t in C" and CQ .
be a continuous
plurisubharmonic function in For e v e r y compact s e t K & fl , t h e r e e x i s t a famlly (n ) of positive i n t e g e r s and a family (g,) of holomorphic functions in SL s u c h that
P
'f
over K. Proof. Let -
=
SUP
l/np
P
log j g p l
denote the function e-'9 extended by z e r o on the complement of l-l and for e v e r y positive number set
8,
Clearly
6,
=
l/h M in(h',,(XS)+). '
is Lipschitz o v e r Cn and
of Chapter I V , - l o g
&
(zI
h is bounded. In view of Proposition 2 W e have
n.
is plurisubharmonic in
SA(s) =
Min (
-, inf &(S)
x
s%Cn
(e-'p(s')
+
Is'-sl x 1) ,
.
For a fixed s in fi , c l e a r l y $ ( s ) where e-'4('') is replaced by 0 when s ' # is the i n c r e a s i n g l i m i t of ZA(s) when 1 tends to zero. W e t h e r e f o r e can uniformly approximate
$
by functions $1 on e v e r y compact s u b s e t K of , we easily obtain Proposition 1.
.-
C o r o l l a r y 1 Let of 12 we have
-
fi
. Applying then
be a pseudoconvex open set in C n ; for e v e r y compact s u b s e t K =
KO(hl)'
A
9
W e have a l r e a d y s e e n that K n is the hull of K with r e s p e c t t o continuous p l u r i s u b harmonic functions in 0 From Proposition 1 , i t is also the hull of K with r e s p e c t to functions l / p l o g l g l when p is a positive i n t e g e r and g a function of and A w e e a s i l y have Ksl = K
-
.
o(n)
ocn).
Using similar methods, we also obtain Proposition 2 . - Let 'p be a continuous plurisubharmonic function on Cn s u c h that eY h a s polynomial growth at infinity. For e v e r y compact s e t K & C n , t h e r e e x i s t a famil~ (n ) of positive i n t e g e r s and a family (q ) of polynomials s u c h that
P
K.
P
'p
= SUP
P
l/n,
1%
lspl
FILTRATIONS
84 We only have t o note that
81
has.
h a s polynomial growth when
of
Corollary 2 . - For e v e r y compact subset K n
A
C", the polynomially convex hull K
K is equal to the hull K of K with r e s p e c t t o plurisubharmonic functions P such that e 'P has polynomial growth at infinity.
'Q
P of 0" cn
7 . 4 . - Approximation theorems with filtration \Ve are a b l e t o s t a t e a new approximation r e s u l t : Theorem 3.- I,et
$
b e a weight function bounded b x 1 0" Cn
and
A' a directed s e t
s.
of ueight functions s u c h that& 8'~dis l a r g e r than some function equivalent to W e assume that > 01 is pseudoconvex for e a c h 8' E A' : the following conditions = j 8 > 01 : a r e equivalent, -n
n
(1)
for e v e r y positive integer N , t h e r e e x i s t s a family (f,)
of functions of
O(Al) so that
- fl , w h e r e k is independent of on
lim N+ (11)
and
N
+M
t h e r e e x i s t a family (n ) of positive i n t e g e r s and a family (g ) of functions
of O(al) so that
P
B
-log
sz;
6
(iii) l / x is the supremum on plurisubharmonic function in some
(iv)
- log $
8
sup l/np
P
of a family
( &)
is plurisubharmonic in
filtration ; (v)
=
is s p e c t r a l for z p~-
log / g p l
(n,), w h e r e e a c h nd %a
ss m &~ ; a t
log-
O(al) is d e n s e in O(&)with
- with filtration. O(Al)
Proof. We only have to follow the proof of Theorem 2 t o show that (i) implies (ii). S e t t i n g 7cd= Ig,\l/n@ in (ii), we immediately obtain (iii). If C)(A') is d e n s e in with filtration t h e r e e x i s t s some positive integer k' s u c h that e v e r y element of
o(8)
belongs t o the c l o s u r e N + k l
-
(?(a') of O(bT) N+klo($)
the norm in .O(s) is l a r g e r than the norm i n N + k l e a s i l y s e e n that (iv) implies (v).
OA ()'
as
in p J + k a S ) , and It is t h e r e f o r e
8 < 1.
Assume that (v) holds. For e v e r y positive i n t e g e r N and e v e r y s E C", one can
..
1;
find functions u,(s) : H u,(s;t), , , un(s) : w u,(s;c) satisfying ( 7 . 2 . 1 ) and ( 7 . 2 . 2 ) . W e e a s i l y obtain ( 7 . 2 . 3 ) that is :
in
N+k
m)
APPROXIMATION THEOREMS
For e v e r y
E E 10, 13
, choose
Eu,(s) in
some
sN+k
I Eu,(s) -
u,(s)l
85
o(n')s u c h that 6
E
.
Thus
13
EEJO,
and condition (i) is proved, a s lim N++m
(cN+ I)'/'
=
lim N++W
c:"
=
I
.
o(g)
W e only have to show that (3(&) is d e n s e in with filtration when (iii) is assumed, It is a consequence of Proposition 2 .of Chapter I and t h e following
8
Lemma 1. - L A b e a locally integrable non negative function on a n open s e t 0 of C". W e consider a sequence , where 0P is a pseudoconvex open s e t of C" containing 0 and
(np,yP) -
- n,,.
u o(ljp) n
P
&
L2(
8: 6 d h )
contains
an)f7
Proof.
W e f i r s t define a sequence s u c h that
2)
$a D N q
n .
8
L2(
g!
L2(
:h2 F
(o( )
9
of
e-%
dX )
d X ) , when
e-% E L:oc
eWfunctions with compact s u p p o r t on
is uniformly bounded independently of q for e v e r y derivative D of
order 1. For instance, i f
p
is a
1;"
function on C" with support in t h e unit ball and s u c h
that dX(s) = 1 and pq(s) = 2 2 n 7 ( 2 q s ) we may take o!q = x q where is t h e c h a r a c t e r i s t i c function of t h e s e t 3 2-'5 Let f b e a function of O(n) fl L2( dX) As dl1(faq) = f d"uq have
Yp(s)
(a).
xq
s ~ 82 .
1sa
.
and t h e right hand side tends to zero when q t e n d s t o infinity. A s the d e c r e a s i n g sequence e - q P
, we
2
can a s s o c i a t e to e a c h q some p
* Pq+l , we
is the limit of =
p(q) s u c h that
FILTRATIONS
86
tends to z e r o when q tends t o infinity. Using H o r m a n d e r ' s l e m m a , w e can find a locally integrable function g on n such that d " g = d" (f Q ) and q p(q) 4 q
st
lgqI2
(7.4.1)
<
dh
e-yP(q)
Eq.
%q) A s d"(f4-
f
9
q
-g
belongs to
)
9
=
= f a - g q is holomorphic in q e-%(q)dh). From ( 7 . 4 . 1 ) we obtain
0, clearly
s!
L2(
6
f
q
5: 8
jgqj2
<
d l
Eq
8: 8
np(q); moreover
9
dX ) . It is easily s e e n that f is the l i m i t of the q d h ). Finally f is the sequence (f ct ) in L2( 8i2 d h ) and thereby in L2( 4q d h ) of the sequence f = f u - g and the proof of Lemma 1 is comlimit in L2( 6, 9 q q plete. so that g tends to z e r o in L2(
8
6
$2
End of the proof of Theorem 3 . Let 0, = when E > 0 is sufficiently small so that 2
b
-log
I ga> 0)
=
t&,
and
. Then
s
ym=
Min (( I/X,)"
, E 2 ),
s u p -1ogy U a
and for e v e r y non negative i n t e g e r N 1 -log
8"
=
Proposition 2 of Chapter IV shows that - l o g is continuous, we can r e p l a c e the family instead of Then
JN'
8.
u (3(n,)n U
s u p - N ' log
xi
(x,)
by a sequence and apply Lemma 1 with
L2(yz'+4d h
O(n)
in L2 ( $ N ' + 4 d h ) contains n L2( L N ' - 2 d h ). We only have now to u s e Proposition 2 of C h a p t e r I . Each
o(
*
ya IS plurisubharmonic in 0, . As i t
o(
o((&)
n I?(
1&N;4
contained in &) and thereby in (3( 54 and Al). F u r t h e r , choosing N ' = 2 N + 2 n + 3 , clearly n L2(8N'-2.d? ) contains whereas IS
O(n)n L2( 8N'+4 d X )
o(n)
is contained in N+3n+7
06).
dX )
.O(x),
liemark. W e only have considered a l g e b r a s and s u b a l g e b r a s . However t h e methbds developed h e r e can be applied to o t h e r c a s e s . For instance Proposition ?.function on
that
Let
be a pseudoconvex open set in C" rfi a plurisubharmonic E of a l l holomorphic functions f Efl such
. T h e vector s p a c e
" 1 ' 4
APPROXIMATION THEOREMS
If\'
e-y
dX
<
87
+M
o(n),when e-'P is locally integrable.
for some positive integer N, is d e n s e i n Proof. Let f -
.
bc an holomorphic function i n 0 and K a compact subset of We want to approximate f by functions of E uniformly on K . We may assume that *,cp 2&]. Clearly the r e ex i s t s some E > O such that K =
1
As each
&Gq is log - plurisubharmonic in y,
'Q +
=
, e ach
(5 F )
2log
Eq
q=o
. Now let
is plurisubharmonic i n 0
S and the s et whcre
8
=
n
&
-log
e-Yl-&
= sup
P
(9P
J
'
(s - E ) + l 2 4-l
d o e s not vanish is exactly the s e t
; obviouslj
0 =
>E f
. Lemma
1
s h o w s that the closure of
u <3(n)
n
L2(s!
e-'P
dX )
P
in L 2 ( $:
$ dX )
contains
0 0 )f i L 2 ( $i2 8 d h ). But JG2 $
is bounded in
G, and the l as t vector space contains f . The statement is t h er ef o r e proved a s d X ) implies compact convergence on K and each convergence in L 2 (
8: 8
o(f),) n L2( 82 e-'PP
dX ) is contained in E
Corollary 1 (L. Hormander).vector space E
Y
Y C Q
Let '4
*
be a plurisubharmonic function on
of a l l e n ti r e functions f such that I l f l ' e-(9
s?
dh
for some positive integer N , is dense i n
<
+GO
o ( C n ) , w h e n e-vis locally integrable.
o(b') is the a lg e b r a of polynomials,
In the particular case when Theorem 3 the following statement :
-
8
Corollary 2 . Let be a weight function bounded by 1 and does not vanish; the following conditions are equivalent :
F
; the
w e deduce f r o m
denote the s e t w h er e
88
FILTRATIONS
for e v e r y positive i n t e g e r N, t h e r e e x i s t s a family (p,) of polynomials So
(i)
that
0" 0 ( r e s p . 0" C")
where
k is independent of N
lim
N++W
(ii) t h e r e e x i s t a family (n B mials so that
0" n ( r e s p . 0" c " ) ;
-log&
c;iN
=
1 ;
of positive i n t e g e r s and a family (q ) of polyno-
P
=
s u p l/n,
B
logIqp\
(iii) l/8 is the supremum on fl (resp. 0" C n ) of a family of l o g - p l u r i s u b h a r monic functions with polynomial growth on Cn ;
(iv)
-log
filtration i n
is plurisubharmonic in
O(S);
is s p e c t r a l for z @
(v)
0 and the polynomials are d e n s e with
m)
with f i l t r a t i o n .
- Polynomially convex open s e t s
7.5.
In Section 6 . 1 . w e have defined polynomial convexity for compact s u b s e t s of Now let
8
b e a weight function on C" s u c h that
86
1 ; we s a y that
8
en.
is
polynomially convex i f 8 s a t i s f i e s the equivalent p r o p e r t i e s (i) to (v) of C o r o l l a r y 2 . An open s u b s e t of Cn is s a i d t o b e polynomially convex i f ,6 is. E v e r y polynois pseudoconvex; moreover, using property (ii) of C o r o l l a r y 2 , we
mially convex obtain Proposition 4
.- E v e r y polynomially convex open s e t is Runge.
T h e c o n v e r s e statement is not t r u e . For instance, an open s u b s e t
C-a polynomially convex (*) . Runge i f and only i f
fL
of C is
is connected. S u c h open s e t s can b e found which are not
T h e infimum of a family of polynomially convex weight functions on Cn is polynomially convex. T h e r e f o r e the i n t e r i o r of t h e intersection of a family of polynomially convex open s u b s e t s of C" is polynomially convex. When 5 is a weight function on A , < 1 , w e denote by S p the smallest polynomially convex weight funcCn such that tion which is l a r g e r than 8 Similarly when fi is a n open set in C", we denote by A the smallest polynomially convex open s u b s e t containing 0 I t is e a s i l y s e e n
5
np
that
.
n p= {(sn); > O} . A
.
a9
POLYNOMIALLYCONVEXOPEN S E T S Proposition 5.-
A
a ) 1/8,
is the supremum of all log-plurisubharmonic functions
with polynomial growth su5h that 7T8 ,C 1 .
b) - l o g is t h e supremum of a l l p log I q \ i n t e g e r and q a polynomial with log ( q -log 8
<
Proof. Obviously -
,
l a r y 2 shows that
b
and
=
Therefore Corollary 1
p is a positive
I
and p r o p e r t y (in) o f Corolexp ( -sup p log q 1) >/ (sup IT )-’ p log \ q \ ). F u r t h e r (sup X )-I 2 i n f ( l / n ) - ,
sp 2 exp(-sup
A
8<
inf ( l / n T is polynom/tally convex. But T C & < 1 implies
8,a 5
.- Let
w be
8, .
>/
and
(l/n)-
.
an open set in C”; a s s u m e that t h e r e e x i s t s a log-plurisub-
harmonic function with polynomial growth X
[G . TX
.
where
TT
s u c h that R < 1 0” w
o is polynomially convex.
Proof. We only have to prove that
(80)~(s) = 0 A
for e v e r y positive integer p , w e have 1/XP(s) w e obtain the r e q u i r e d property.
for e v e r y s E
and n > 1
[G . A s n p &
> (za)&). When
0”
$ 1
p tends t o infinity,
Let f o r i n s t a n c e p l , . . . , p , bepolynomials. T h e o p e n s e t {Ipll + . . . + J p , ( 4 1 1 or the open polyedron \ p l \< 1 , . . . , (p, I< 1 , are polynomially convex. A s e v e r y polynomially convex compact s e t h a s a fundamental system of neighbourhoods composed of polynomial polyedrons, we obtain Corollary 2 . - Every polynomially compact s u b s e t of Cn h a s a fundamental system of neighbourhoods composed of polynomially convex open sets, T h e converse statement is t r u e ; if K is the intersection of some PolynomiallyconA vex open s e t s , we e a s i l y see that K p c K . But Corollary 2 of S e c t i o n 7 . 3 s h o w s A A that Kp = K p and K is t h e r e f o r e polynomially convex. 0
In p a r t i c u l a r , the i n t e r i o r K of e v e r y polynomially convex compact s e t is polynomially convex. T h e r e e x i s t , however, polynomially convex open s u b s e t s
are not bounded or s u c h that
fi
Notes
(*) For instance t h e open s u b s e t of C defined by convex i f and only if d >, 1; writing z = p eie with s u b s e t defined by
1
n of
C which
is compact but not polynomially convex (*).
+ exp ( - l/@fi
y
< exp ( - 1x1 d, 0 , 0 <\$\S ?K
is polynomially
, the open
is polynomially convex i f and only if d > l ;
[ O , 1 [ is not polynomially convex. Such p r o p e r t i e s
the complement in the unit d i s c of can be verified by using methods of t h e author (2).
90
FILTRATIONS
Filtrations have been introduced by L . Waelbroeck (?) i n his sp ect r al theory of b a lg eb r as ; h e r e the word is used with a more r e s tr i ct i v e meaning. Proposition 1 h as been stated by H . J . Bremermann (2) (see also P. Lelong ( I ) ) , but h i s proof is not c o r r e c t . Theorem 2 is a n improvement of the same r esu l t , as the p r o p er t i es are valid in the whole open set ; equivalence between condition (i)and a condition similar to (iii) h as also been obtained by N . Sibony by means of methods of H ar t o g s : the su p r emum of l/ng log /gB/ is only replaced by its lowest upper semi-continuous majorant. The technique of Lemma 1 is due to B.A. T a y lo r ; it h as been used in this context by
N . Sibony. The concept of a polynomially convex open s e t is a refinement of convexity with r es p ect to introduced in Chapter V ; polynomially convex open sets are So)-convex and thereby Runge. T h e author d oes not know whether convex open sets which are not polynomially convex ex i st ,
o(
o(go)
o(so)-
91 BIBLdOGRAF’HY
1 A r e n s , R. and C a l d e r o n , A. P , ( ) Analytic functions of s e v e r a l Banach a l g e b r a elements, Ann. of Math. 62 (1955), 204-216. Bochner, S. and Martin, W.T.
1 ( ) Functions of s e v e r a l complex v a r i a b l e s . - P r i n -
ceton University Press, 1948. Bourbaki, N.
( 1) T h k o r i e s s p c c t r a l e s , c h a p i t r e 1 , Algebres norm6es.- Paris,
Hermann, 1967. 1 Brcmermann, H J . ( ) Uber d i e Aquivalenz d e r pseudokonvexen C e b i e t e und d e r Holomorphiegebiete i m Raum von n komplexen Veriindlichen, Math. Ann. 1 3
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