Domains of Existence in Infinite Dimension

Infinite Dimensional Holomorphy and Applications, Matos (ed.) 0 North-Holland Publishing Company, 1977

DOMAINS OF EXISTENCE IN INFINITE DIMENSION

By G E T U L I O KATZ

INTRODUCTION Among the subjects studied in infinite dimensional

holo-

morphy the one of analytic continuation is nerhaps the most develoned. (See 161 for a survey and a fairly comDlete granhy)

bihlio-

.

The Levi problem asks whether it is true that a domain is pseudo-convex if and only if it is a domain of existence.In the first nart of this paper we rJive a positive answer to the

Levi

problem in the case of Riemann donains over Banach spaces with Banach apnroximation pronerty (R.A.P.).

This result

follows

from investiqations about nroperties of permanence for

domains

of existence under elementary set operations.

P

In the second part we m o v e that a convex set in (A

any index set) is a domain of existence. This result

ses that the behavior of the space of polynomials is

n

(A)

stres-

directly

related to the answer to the Levi problem. This naper is hased an the author's doctoral thesis written under the quidance of Leonoldo Nachhin at the University of 239

240

KATZ

G.

Rochester. 1.

Let E and F be locally convex

Hansdorff

spaces (1.c.s.);

U C E be an open set: @cmE;F) the set of all continuous m-ho-

of

moqeneous polynomials from E into F; C . S . ( F ) the set continuous seminorms in F. Denote by %(U;F)

all

the set of holomor

phic mappings from U into F as defined below. A {unction f

DEFINITION

5

E

all?

:

.

U

+

F

h o l o m o k p h i c i i $oh

i h

U a n d m = 0 1 1 1 2 1 . . t h e h e e ~ i Pm ~ Et @(mE;F) (F) t h e h e

f3EC.S.

lim P[f(x) m+m DEFINITION

-

m

c

n=0

:

P,(X-S)J

X * E

i h

X

#

o

$,

uniformly on V.

a -!ocn-! h o m e o m o h p h i h m .

over the same basic space; da

(where A

a E

C.S.(E);

C $$(XI

a

X LA a C o n f l Q c t e d H a u h d o h 4 b bpace

Holomorphic functions; morphism between Riemann

tively to

(OR

,that

nuch

=

nuch t h a t

t h e paih (X,p) i n c a l l e d

L e t E b e a l.c.s.,

R i e m a n n d o m a i n o v e h E id and p

u

exiht.4 V(E)

all

:

X

+

+ IR

domains

the distance in X rela-

The concept of A-domain of

holomorphy

1 : etc. I are all defined in the obvious way.

Let (Xllpl)and (Xz1p2)be two Riemann domains over and E2 respectively. Let X = X 1XX 2 and p = (pl1p2) : X then (X,p) is a Riemann domain over E1xE2.

+

El E1xE2'

Assume that Ell E2 are metrizable 1.c.s. whose topologies and 8, < p 2 respec are generated by seminorms al < a2

...

...

tively. Then E = E1XE2 is a metrizable 1.c.s. whose topology is

. . where

yi (x,,~,) = sup ( a i (x1 I Pi (x,) 1 . X2; Y = X x(x2); 9: Y El .Then ( Y l q ) is 1 (Xl1X2)+ P1(X1)

generated by y1
E

a Riemann domain over El such that Y

-+

=

X1.

DOMAINS OF EXISTENCE

Let (Xi,pi)

THEOREM 1

241

a n d (Y,q) ah a b o v e .

i = 1,2

ifi

Then

( x , p ) i 6 a domain o f i e . x i 6 t e n c e 6 0 i d ( Y , q ) . W e u s e some r e s u l t s o f 151 and a l l d e f i n i t i o n s r e a u i r e d

PROOF

i n t h e proof c a n be found i n t h i s p a p e r . Suppose (X,p) i s a domain of e x i s t e n c e . Then by

4.3 o f [5!,

i s an

t h e r e i s a n a d m i s s i b l e c o v e r i n g v o f X I such t h a t X

A--

domain o f holomorphy.

Let

Q. =

v

W e claim t h a t

t h a t Y is A

1 v

(V f ) Y

E

V

a is admissible.

have t h a t X i s AV

'v may

and A U

by f l ( y ) = f ( y ) . As f E A v

So

I lfll l v n y

f (u)

#

51 I f l [

f ( w ) = f l (w)

Iv

<

m;

+

-convex,

I If1 I v

Again by theorem 3.6

%(Y)

<

m

defined

V

for a l l

v.

E

and f l ( u ) =

Y i s AZI - s e p a r a t e d . t h e n by theorem 3.6

.,

we

o f 15J

i n X s u c h t h a t dim ( x n ) + 0 f o r

s u c h t h a t s u p l f (xn) I =

Y s u c h t h a t dam (Y,)

Y

0 f o r a l l Y,.

v -separated,there

it follows t h a t f l E A

a l l yml t h e r e e x i s t s f E Av

dXm(Yn) Y

As X i s A

f(w). Let fl E

then

have t h a t f o r a l l s e q u e n c e s (x,)

L e t (Y,)E

prove

W e shall

-separated.

w.

. Therefore

Now as X i s A

we

-separated.

such t h a t f ( u ) #

E AV

of Y.

a n open c o v e r i n g

B y p r o p o s i t i o n 4.2 o f [5]

-convex and A v

u -convex

be t a k e n c o u n t a b l e .

3 is

1.

Assume f u , w ) C Y C X , u #

exists f

theorem

+

0

for

W e have t h a t f l E A

of [5] w e g e t t h a t Y i s A

m.

all

am.

Then

and s u p ! f l ( Y n )

u -convex

I*. and

by p r o p o s i t i o n 4.2 and theorem 4.3 t h a t Y i s a domain o f e x i s t -

...

ence. q e d

DEFINITION

A Banach 6 p U C e E i 6 s a i d t o h a v e t h e Banach a p p h o x

i m a t i o n p R o p e h t q (8.A.P.) i.( E i b b e p a h a b e e and R h e R e . 6eque.nce 0 4 o p e h a t o m 0 4 f i i n i t e doh

ale x

E E.

Rank

( u n I n c-

exibt6

a

6 u c h t h a t un(x)+ x

242

G.

KATZ

L e t (Xl,Pl) b e a Riemann domain o u e h a Ranach Apace

COROLLARY

El w i t h B.A.P. T h w X1

X1 i n a

pneudo-convex i4 and o n t ! j id

i d

domain O X e x i n t e n c e . PROOF

It is known that a separable Banach space has B.A.P. if

and only if it is a direct subspace of a Banach space with

ba-

sis. (see C4-J). Let E = El

G where E is a separable Banach space

x

with

basis. Assume (X,p) is a R i m domain over E such that X=X1x G and p = (pl, id): X * E. As X1 and fX,P) * Now by

[l]

so

are pseudo-convex

G

is

(X,p) is a domain of existence, hence by the-

orem 1, X1 x { O ] is also a domain of existence.

Finally

as

X1x{O)=X1 it follows that (Xl,p$ is a domain of existence.q.e.d. In contrast with theorem 1 where separability was crucial, we shall state. THEOREM 2

l.c.6.

~ e it = 1,2,

Ei. Then

x

a2 i n

a domain o h e x i b t e n c e

Suppohe E i d a

LEMMA 1

i b t e n c e i( and E

k!.c.b.

BC'U), nuch t a h t ( o h neighboahood w (z-bak'anced

aU t h e h e doen n o t e x i n t a

LEMMA 2

Suppobe E and

G

PROOF OF THE THEOREM x

6E

%(Wl

2,

E 2 ) and f2 E

4 agheeb

and

iuith

0 4 w flu and z E aU fl a R t

ahe t w o t . c . n . ,

then

'u x

G ib

a

do-

E i n a domain 0 4 e x i n t e n c e .

main 0 4 e x i n t e n c e phouided

%(%1

i b a domain 06 ew

o n l i t i{ t h e h e e x i d t o 9 E

g i n a c o n n e c t e d component R

E

eXibtenCe.

and UCE.

and c o n v e x ) buch t h a t t h e h e e x i b t n

fl

06

a domain

a

in

The proof is based on two technical lemmas.

PROOF

att Z

q,

q i c E~

x

(x1

U 2= %(El

x

x

2,)

E ~n )

(

E~ xQ2).Let

such that:

'u1 x EZ

DOMAINS OF EXISTENCE is the domain of existence of fl and El

243

u 2the domain of

x

ex-

istence of f2.

a(%,

We have that (

3,

and x E Aopenc El fl

+

f2

I (2' 1,

E

A, then A n ( a 3 ,

x

a 2 )has

x

If not, there $(

such that

)

5

a ) Suppose Z E

a(%,)

21,.

x

+

b) Suppose Z E

c) If

z

f2 in

a,

(aU,)

E

claimthat

as natural domain.

u,)

I

w ( Z ) and

-

9

x

of

domain

i

x

wlC vlc-Q1

E2.

9 . Let 0 be a connected CXXJ

Q,).

x

-

8;

a%,.

x

E

B2CQ2'

f2

E

% ( wl) and

in 8. This contradicts the fact that fl has of existence.

We

3 2 ,

Then there exists

Wlc W, 0 (21, = fl

ff.

x

a(n).

E

Wl#

We have

#

a ?ll

E

fl + f2 in a connected component R

domain El' such that Z E B1 x B2 = B1 Besides as 2 E a ( R ) then D f l

n 0

a ,)

X

21, x 21, E ?(qlx

exists Z

Wn('?J, x U2)such that z

ponent of

u caql x Q.,N

aq,)

x

It i s easy to prove that for all x

3'21,).

x

(aul

21,) =

x

2,

x

9- f2=fl

E2 as

domain

Same proof as in (a).

(aQ2), let

W(Z) as in

lemma 1. sup

pose A1 and A 2 convex, Zi - equilibrated oDen neiihborhoods of Ai and Z E A1 X A 2 C . w ' , where Z = (ZlIZ2). As Z E a(Q),there exists (xlIx,)

E

R f) (A1

A,).

x

So (ZlIx2) E A1xA2; s=(Z1,x2)

E

ul) q,. Let A3 = connected component of %, 0 such that x2 Then A1 = v,;wl w ;vlC:EIX?$; wl '&, E2 and (xl,x2) R fl (A1 Thus reasoning as (2

X

A,

E A3.

C

X

s E

X

A3

E

in Dart (a) we would cJet that

X

a,

existence of fl. It follows that

X

21,

A2).

E2 is not the domain X

of

(1L2 is a domain of ex-

istence. q.e.d.

2.

In this section we shall follow the techniques of 121

to

244

G. KATZ

prove that in .f? ( A ) the open convex sets are domains of F tence. Let F be a Banach space,

F containinq the oriain and Let c > 0 such that

ac F be a convex open subset of

the closed unit ball in F.

Efi

C-U ; usinq the onen-mappinq

Hahn-Banach theorems, one can nrove that for all x exists $x

E

F' satisfyinq:

I lGxl

Re($,(Z))

<

1 whenever Z

E

U.

(Z) =

1

Let

f

X

have that fx

have t h a t

PROOF*

1 < 2 / ~ , Re $,(x)

, where i

(z)-$x

Fofi ale Z E U , y E ( 0 , sup

XEaU

Let Z

E U

03)

and fix Y

a = sup (r>OlrZEU) and Vz

,Y

E

= (1/2 (1

-

5 Re (4,(Z))

$,(x))

+

5

I 7

a ) . So

U.

+ ~vRCU,

+

2 y B c U . kfine

If

YEy

E

;

(1

-

Re(bx(Z)) - 1 = IRe($x(Z+Y)

-

lfx(Z+Y) 1 5 M Z I y f o r all YEyR. q.e.d.

cpxfx) I 2 1/2 (1- I Let .f?,(A) , p

Z E

(l-Re($x(Z)).Therefore

-i 1 (Re($x(Z))-l). But Re(Gx(Z)) 5 1. , therefore

-

We

m.

-

Re($x(Z+2Y)-@x(x)) 5 0, then Re $ ( Y ) 5 Re($x(Z+Y)

N is fixed.

E

such that Z

(Olm)

and

= 1

duch t h a t Z

(fx(Z+Y)I 5- M Z I y <

sup YEYB

and there

E aU

# 0 for a l l

$,(x))

since Re($,(Z)-

%(U)

E

PROPOSITION 1 We

[a,

exis-

c

be the set E(A)=(f:A+Cl

E[l,m),

endowed with the norm 1 If

I1

Banach space.Let

given by

= ( Z If ( a ) aEA

1)' 'ID

I a (PI =

E

a€ A fp(A)

If

(a)lp<-l

is

a

1 if a = f?

0 otherwise

THEOREM 3

Let 0

E U

c t

P

(A)

b e an o p e n coyluex n e t . T h e n

U i n

__

*

This elegant proof is due to David Prill, nine was consider-

ably more complicated.

245

DOMAINS OF EXlSTENCE

0 4 existence.

u domain

If U =

PROOF

eP ( A )

the result is clear. So we may

U # l p ( A ) , and we may also assume that A is well

consider

orderedl

say

A = [O1$) where $ is the cardinal number of A. Finally it isp? sible to prove that there exists a dense subset (xnlaEA in aU. The proof of the theorem is based in three claims. T h e J w e x i b t d C:A+A ouch t h a t 1

CLAIM 1

1

injective

i d

= 0 if{; where CJ e ( P I is the (xY)l(P) - W P ) associated family of coefficients functionals.

and

corresponding

'1

? if p is integer,

Let

q

=

(p]

+ 1 otherwise, where

c

pI = integer part of p \

Z ) , c a n n o t be continued .f t(a) xu 0ue.4 xa. ( I n t h e d e ; ( i n i t i o n O X f t . . k e i = CJ + 1). xa C Ca Z:(a). f ( 2 ) where C a is defined as Set q ( Z ) = aEA a' f01lows

The .(unction Z

CLAIM 2

+

2'

.

We shall conclude the proof showing that q cannot be con-

tinued over any point xal

aEA;

and therefore g cannot be

tinued over any point of a U since {xn} deed,

J:

Y'P such that

.

a EA

is dense in aU.

C ( S X ~ + ~ L ~ ( ~ ) ) 'fxy(Sxy+ylt(B)) = 0

conIn-

for all 5 , y E C

Exg+yLZ(p) E U, because the way we defined 1 . Now if

G. KATZ

246

as r

+

0.

If as r

+

0

on a sequence of points with infimum 0. Indeed in this

case

= 1 and the behavior of the sum is qiven by the last term, P which goes to fininity by the claim 2. This finishes the proof.

C

BIBLIOGRAPHY

111 HERVIER, Y., Sur la problgme de Lev1 pour les e s p a c e s h G s Banachiques, C.R. W a d . SCI, V.275, ser A.p.

821,

1972. 121 JOSEFSON, B., Counterexample in the Levi problem.

Proce-

edinqs on infinite dimensional holomophy, Lecture no tes in Mathematics. Snringer Verlag, V. 364,

1973,

168-177. 1 3 ) NACHBIN, L., Topology on spaces

of holomornhic

maminqs.

Erqebnisse der Mathematik und ihere qrenzze

biete,

Sprinaer-Verlag, Hejt-47, 1969. 141 PELCZINSKI, A , , Any separable. Banach space with the bounded approximation pronerty is a comnlemented subspace of

DOMAINS OF EXISTENCE

247

a Banach space with basis. Studia Math: E 40,

1971,

p. 239-243. [5] SCHOTTENLOHER, M.,

Analytic continuation and regular clas-

ses in 1.c. Hausdorff snaces. Portuqaliae Matematica. (to appear). [6] SCHOTTENLOHER, M.,

Riemann domains: Basic results and open

problems. Proceedings on infinite dimensional holomornhy. Lecture notes in Math, Springer-Verlaq V.364, 1973.

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