Spectral theory and complex analysis

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

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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


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(&)


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; 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


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&)


, 5 J(s) for in B, the ball B is contained in 18 >O] and from subharmonicity of If Ip, we get


,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 ?/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"

,


$a,+, 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 + 2O 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,+ - 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(

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,
/ 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
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 ) : ( 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


+. . .+ (-l)r+l< z - s ,

(z.-s)r->

f(?

+



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



. 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 = 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
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,
0, and that t is such that weight function bounded by A s jTa I/y.(, obviously

is a


ESP,

wealso have


&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

.

- 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
/ 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\

,


2 - s , s"

if

sv.

&/2)-No

and using 1

2-s,

4

0

< 2 I ).

and also 'u,

are holomorphic in




is. Moreover

.

is bounded, we have

~(S)-~(S II

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 ,
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

(1965), 89-152. 2 ( ) An introduction to complex analysis in several variables.- New-

York, D. Van Nostand Company, 1966. (3) Generators for some rings of analytic functions, Bull. A m e r .

Math. SOC.2 (1967), 943-949. 1 Houzel, C (editor) ( ) Skminaire Banach(mimeographed), Ecole Normale Supkrieure,

.

P a r i s , 1963. Kelleher , J . J. and Taylor, B. A , ( I ) Finitely generated ideals in rings of analytic functions, Math. Ann. 1 3 (1971), 225-237. 1 Kiselman, C.O. ( ) On entire functions of exponential type and indicators of analytic functionals, Acta Math. lL7 (1967), 1-35. 1 Lelong, P . ( ) Fonctionnelles analytiques e t fonctions entibres (n variables), Seminaire de Mathdmatiques supdrieures Montrgal, Presses de I'Universit6,

s.-

1968. 1 Leray, J . ( ) Fonction de variables complexes: sa reprbsentation comme somme de

puissances nkgatives de fonctions lingaires, R. C. Acad. Lincei, (8) 20 (1956), 589-590. 1

Malgrange, B. ( ) S u r les systhmes diffgrentiels h coefficients constants, Coll. Int. du C.N.R.S. 5 7 (1963), 113-122. 1 Martineau, A. ( ) Indicatrices de croissance des fonctions entieres de n variables, Invent. Math. 2(1966), 81-86. 1

Narashiman, R , ( ) Cohomology with bounds on complex spaces, Several complex variables, Lecture Notes in Mathematics 1 5 5 , 141-150.- Springer Verlag, 1970. 1 Norguet, F ( ) S u r les domaines d'holomorphie des fonctions uniformes de plusieurs

.

variables complexes, Bull, SOC. Math. France Oka, K .

1

( 1 954), 137-1 59.

( ) S u r les fonctions de plusieurs variables I , J. S c . Hiroshima Univ.

(1936), 245-255.

6

93

2 ( ) S u r les fonctions de plusieurs variables I X , Jap. J. Math.

a (1953),

97-155. 1 Poly, J.B. ( ) dll-cohomologie 3 croissance, S6m. P. Lelong 1967-68, Lecture Notes in Mathematics 1 2 , 72-80.- Berlin, Springer Verlag, 1968, 1 Rubel, L. and Taylor, B.A. ( ) A F o u r i e r series method for entire and meromor-

.

phic functions, Bull. SOCMath. France 96 (?968), 53-96. I Shilov, G.E. ( ) On decomposition of a commutative normed ring in a direct sum of

ideals, Amer. Math. SOC.TransL 1_(1955), 37-48. 1 Sibony, N. ( ) Approximation pond6ri.e des fonctions holomorphes dans un ouvert de C", Skminaire Choquet 1970/71, no 21, 14 p. 1 Skoda, H. ( ) Systeme fini ou infini de g6n6rateurs dans un espace de fonctions holornorphes avec poids, C. R . Acad. Sci. Paris A 273 (19711, 389-392. 1 Taylor, B .A. ( On weighted polynomial approximation of entire functions, Pacific J. Math. 2 (19711, 523-539. 1 Waelbroeck, L. ( ) Etude spectrale d e s algebres compl&tes, Acad. Roy, Belg. C1. S c i . M h . , 1960. (2) Lectures in spectral theory, Dep. of Math., Yale Univ., 1963. ( 3) About a spectral theorem, Function algebras (edit. by F Birtel),

.

310-321.-

-

Scott, Foresman and Co, 1965. (4) Some theorems about bounded structures, J. Functional Analysis

? (1967), 392-408.

( 5) Topological vector spaces and algebras, Lecture Notes in Mathe-

-

.

matics 230. Berlin, Springer Verlag, 1971 1 W e i l , A. ( ) L'intkgrale de Cauchy et les fonctions de plusieurs variables, Math. Ann.

111 (1935).

1 Whitney, H. ( ) On analytic extensions of differentiable functions defined on closed

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(1948), 635-