Studies on the Levi Problem and the Indicator of Entire Functions

九州大学学術情報リポジトリ

Kyushu University Institutional Repository

無限次元空間におけるレビの問題と指数型整関数の 指標に関する研究

西原, 賢

https://doi.org/10.11501/3070071

出版情報：Kyushu University, 1993, 博士（理学）, 論文博士 バージョン：

権利関係：

\

Studies on the Levi Problem and the Indicator of Entire Functions

in Infinite Dimesional Spaces

By

Masaru NISHIHARA

Fukuoka Institute of Technology

Contents

Preface

Part I ^{· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·. · · · .. . ·}1 On a pseudoconvex domain spread over a complex projective space induced from a complex Banach space with a Schauder basis,

(Journal of the Mathematical society of Japan, vol.39, No.4, pp.701-717, October, 1987.)

Part II ^{· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·} 27 On the indicator of growth of entire functions of exponential type

in infinite dimensional spaces and the Levi problem in infinite dimensional projective spaces,

(to appear in Portugaliae Mathematic a.

Preface

Let E be a complex locally convex Hausdorrf space and f be an entire function

of exponential type. Then the indicator I1 of the entire function f is the function on E with values in [ -oo, oo) defined by

If(z) ⁼ lim sup lim sup -log 1 !f(tz')l

z1-+z t-+oo i

for every z ^E E. The indicator has the following properties.

(1) If is plurisubharmonic.

(2) If is positively homogeneous of order 1, that is, If(tz) ⁼ tif(z) for every

t ^{> 0 and} z ^E E

Conversely when given a plurisubharmonic function p on E which is positively

homogeneou of order 1, we consider the problem to ask whether there exists an entire function f of exponential type on E with If ⁼ p. Kiselman([18] in Part II), Lelong([19] in Part II) and Martineau([22] in Part II) solved affirmatively this problem in the case of E = en. Their results are called the indicator theorem of entire functions of exponential type on en.

In the present thesis we investigate first the Levi problem in infinite dimensional projective spaces, and then apply this result to the research of the indicator of entire functions of of exponential type in infinite dimensional spaces. The main theorems are the following two theorems.

THEOREM 1. Let E be a separable Frechet space with the bounded approximation property or a DFN-space and (^w VJ) be a Riemann domain over the complex pro- jective space P(E) induced from E. Assume that w is not homeomorphic to P(E)

through VJ. Then the following statements (1), (2), (3), (4) and (5) are equivalent.

Moreover if w is an open subset of P(E), the statements (1), (2), (3), (4), (5) and (6) are equivalent.

(1) w is pseudoconvex.

(2) For any finite dimensional subspace F of E, <p-1 ₍ P( E)) is a Stein manifold.

(3) w is a domain of holomorphy.

( 4) w is a domain of holomorphy and holomorphically separated.

(5) w is a domain of existence.

(6) There exists a non-constant holomorphic function f on ^w such that, for

every connected open neighborhood V of an arbitrary point on the boundary of ^w, each component of ^w n V contains zero off of arbitrarily high order.

THEOREM 2. _Let E be a separable Frechet space with the bounded approximation property or a DFN-space, and p be a plurisubharmonic function in E which is posi- tively homogeneous of order 1. Then there exists an entire function f of exponential type on E _{such that}

for every z E E.

P(z) = limsuplimsup �log lf(tz')l

z1-+z t-+oo i

CoROLLARY 3. _If E is a nuclear Frechet space with the bounded approximation

property or a DFN-space, ther exists an analytic functional p on the strong dual space E' of the space E such that

for every z E E.

p(z) ⁼ limsuplimsup �log lp(exp(tz'))l

z1-+z t-+oo i

In order to prove Theorem 1 we first prove by using the n1ethod of U eda( [43]

in Part II) that a pseudoconvex Riemann domain ( w, <p) over the projective space

P(E) is the quotient space, by C*, of a pseudoconvex Riemann domain (S1, <I>) _with

C* -action over E. _{And then by} researching invariant properties, with C* -action,

Ill

of the Riemann domain (n, <I>) over E, we do those of the Riemann domain (w, <p)

over P(E). Last, we prove the theore1n of Cartan-Thullen's type in a Riemann domain over a projective space induced fro1n a separable metrizable locally convex space, and then prove that a pseudoconvex Riemann domain in Theorem 1 satisfies the assumption of the Theorem of Cartan-Thullen's type. Thus we complete the proof of Theorem 1. The proof of Theorem 2 is based on the characterization of pseudoconvex domains of the projective space P(E) in Theoren1 1. We use, for the proof of Corollary 3, the bijection between the spa.ce of all entire functions of exponential type on E and the space of all analytic functionals on the dual space E' by the Fourier-Bore! transformation.

The author would like to express his sincere gratitude to Professor Joji Kajiwara for his continuous encouragement and valuable suggestions.

Part I

On a pseudoconvex domain

spread over a complex projective space induced from a complex

Banach space with a Schauder basis

Introduction.

Oka(18] solved the Levi problem, which is the problem to ask if a pseudoconvex domain is a domain of holomorphy, in a domain spread over en. At the same time, Bremermann(l] and Norguet(16] solved this problem in en. Their results were extended to a domain spread over the complex projective space P n( C) of dimension ⁿ by Fujita[4], Kiselman[9] and Takeuchi[22].

In the last fifteen years, the Levi problem has been discussed in various infinite dimensional spaces. Gruman[5] and Gruman and Kiselman[6] solved this problem in a complex Banach space E with a Schauder basis, and Hervier[7] extended this result to a domain spread over E. Dineen[2] and Gruman[5) solved this problem in an infinite dimensional vector space E with the finite open topology, and Kajiwara[8) extended this result to a dotnain of the complex projective space induced from E.

The aim of this paper is to prove the following two theoren1s having their sources 1

in the Levi problem and in the imbedding theorem of a Stein manifold.

THEOREM 1. Let E be a complex Banach space with a Schauder basis, and

P(E) the complex projective space induced from E. Let (0, ¢) be a domain spread over the complex projective space P(E). Suppose that 0 is not homeomorphic to P(E) through ¢. Then the following conditions are equivalent:

(1) ⁿ is pseudoconvex.

(2) For every finite dimensional linear subspace F of E and the projective space P(F) induced from F , the inverse image ¢-1 (P(F)) of P(F) by ¢ is a Stein manifold.

(3) ⁿ is a domain of holomorphy.

( 4) ⁿ is a domain of existence.

THEOREM 2. Let ^H be a separable complex Hilbert space, { e1 }f=1 an orthonor­

mal basis of H, and P( H) the complex projective space induced from H. Let ( 0, ¢) be a pseudoconvex domain spread over P(H). Suppose that 0 is not homeomorphic

to P(H) through¢. We denote by Hn the linear span of the set {e1, e2, ···,en} and denote by P(Hn) the complex projective space induced from Hn. Then there exists an injective holomorphic mapping f of 0 into H such that for every positive integer

n the restriction mapping fl¢-1(P(Hn)) off on ¢-1(P(Hn)) is a regular and proper holomorphic mapping of ¢-1 ( P( H n)) into H.

The author would like to thank the referees for their kindly advice, valuable suggestion and encouragement.

1 Banach complex manifolds and domains spread over Banach complex manifolds.

Let E and F be complex Banach spaces, and U an open subset of E. ^A mapping

f : U ^{� F} is said to be holomorphic in U if f is continuous in U and if, for any (a, b) E U ^x (E-{0}) and for any continuous linear functional ^a E F', the composite mapping A� a o f(a + Ab) (A E C) is holomorphic where it is defined. A function

p : U � [ -oo, +oo) is said to be plurisubharmonic if p is upper semicontinuous in U and if, for any point (a, b) of U ^x (E-{0}), the function A� p(a + Ab) (A E C) 1s ubharmonic where it is defined .

A Hau dorff space M is called a complex manifold modeled on a complex Banach space E if there exit a family :F = {(Ui, ¢i)^;i E I} of pairs (Ui,¢i) of open sets Ui of M and homeomorphisms ¢i of open sets Ui onto open sets of E satisfying the following conditions:

(1) For any elements i, j of I with UinUJ =f ^0, the mapping ¢io¢;1 : ¢J(UinUJ) ^�

¢i( Ui n UJ) between open sets in E is holomorphic.

(2) uiE!ui ⁼ M.

:F is called the atlas of M. An element of F is called a chart of M.

Let M and N be complex manifolds with atlases { ( Ui, ¢i); i E I} and { ( U �, ¢�); ^a E

A} respectively. Then a mapping f : M � N is said to be holomorphic if, for any

i E I and a E A with f(Ui) n U� =f ^0, the mapping ¢� o f o ¢i1 is holomorphic.

Particularly, if N = C, f is called a holomorphic function. We denote by H(M) the family of all holomorphic functions in M. A function p : M � [ -oo, +oo) i said to be plurisubharmonic if, for any i E I, the function f o ¢i1 is plurisubharmonic.

We consider subsets �1 and �2 in C2 defined by

3

A complex manifold ^M is said to be satisfy the ^J( ontinuitatssatz if any holomorphic mapping of a neighborhood of .61 into ^M is extended holomorphically to .62.

Let ^M be a complex manifold. If there exists a local biholomorphic mapping ¢ of a complex manifold 0 into ^M, (0, ¢) is called a region sprea over ^M. Moreover, if 0 is connected, (0, ¢) is called a domain spread over ^M.

Let ( 0, ¢) and ( 0', ¢') be regions spread over ^M. If a holomorphic mapping ). of 0 into 0' satisfies ¢ = ¢' 0 A, ). is called a mapping of (0.¢) into (0', ¢'). If (0', ¢')

is a region spread over ^M, then a mapping ). of (0, ¢) into (0', ¢') is said to be an :F-extension of 0 if for each f ^E :F there exists a unique f' ^E H (0') such that f' ^o). =f. ^A mapping). of (0, ¢) into (^w', ¢') is said to be a holomorphic extension of 0 if). is an H(O)-extension of 0. 0 is said to be an :F-domain of holomorphy if each :F-exten ion of 0 i an isomorphism. 0 is aid to be a domain of holomorphy if 0 i an H(O)-domain of holomorphy. 0 is said to be a domain of existence if there

exists f ^E H(O) such that 0 is an {!}-domain of holomorphy.

Let ^E be a complex Banach space with a norm II · II and let (0, ¢) be a region spread over ^E. For a point z of ^E and for a positive number E, we define the open ball ^B ( ^{z, t}) by

( 1.3) B(z, E)= { ^{wE E ;} II ^w- z II< E}.

For any point x of 0, there exists a positive number E( x) such that, for any positive number ^f with ^f < E(x) there exists uniquely an open neighborhood 6(x, E) of x which is mapped by¢ homeomorphically onto the open ball B(¢(x), E). The open neighborhood .6( x, ^f) is called the open ball in 0 with center x and with radius ^f.

We define the boundary distance function dn(x) on 0 by

(1.4) dn(x) = sup{x; the open ball 6(x, E) exists}.

Let a and b be points of n. By a line segment [a, b] inn we mean a set inn containing the points a and b and homeomorphic under ¢ to the line segment [¢(a), ¢( b) J in E.

By a polygonal line [x0, X1, ^{· · ·, xn}] in n we mean a finite union of line segments of the form [x1_1, x1] with j ⁼ 1, · · ·, n.

REMARK 1.1 Let x and y be two points which belong to a connected component

of n. Since there exists a polygonal line [ ^Xo, xl' ^{. . . 'Xn}] with ^{Xo = X} and with ^{Xn =} y ^I

there exists a finite dimensional linear subspace F of ^E such that the set { x, y} is contained in a connected component of the inverse image ¢-1 (F) ofF by ¢ .

2 Complex projective spaces induced from complex Banach spaces.

In this section we first give some properties of a complex projective space induced from a complex Banach pace. Then we give the definition of pseudoconvexity of a domain pread over the complex projective, and prove some lemmas with respect to pseudoconvexity.

Let E be a complex Banach space with the norm II · II· _{Let z and z'} be points in E - {0}. ^z and ^z' are said to be equivalent if there exists a complex number ). E C- {0} such that z' = AZ. The quotient space P(E) _of E- {0} by thi

equivalence relation is called the complex proJective space induced from ^{E. We}

denote by Q the quotient map of E- {0} onto P(E). _{For any} � ^{E E -} {0}, we denote by [�] the equivalence class of ( Then we have Q( �) ⁼ [ �].

Let E' be the complex Banach space of continuous linear functionals on E. _We

set

(2.1) 5 = { ( ^f, a) ^{E E' x E; f} (a) ^f 0}.

5

-

For each f ^E E' -{0}, we consider a hyperplane E(f) of E and an open subset U(f) of P(E) defined by

(2.2) ^{E(f) = {� E E; f(�) = 0},}

(2.3) U(f) = {[�] E P(E); f(�)-# 0}

respectively. For every (f, ^a) ^E S, we define a homeomorphism cP(f,a) of U(f) onto E(f) by

cP(f,a)([�]) ⁼ (1/ f(�))�- (1/ f(^a))^a

for every[�] E U(f). The family {U(f), cP(f,a)}(f,a)ES defines the complex structure of the projective space P(E).

Let S(E) be the unit sphere in E. Then the topological space P(E) is a quotient space of S(E). The topology of S(E) ^as a subspace of E induces the topology on the quotient space P(E). S(E) is a principal fibre bundle over P(E) with circle group. Since S(E) is a subspace of the metric space E, the metric on S(E) induces a metric d(,) on P( E) by

(2.4) d(p,p') ⁼ inf{ll ^{z- z'} II; ^{z E} Q-1(p) ⁿ S(E), ^{z' E} Q-1(p') ⁿ S(E)}

for any points p and p' of P(E). Since E is complete and S(E) is closed, S(E) is a complete metric space. From the compactness of the fibre of S(E), it follows that P(E) is also complete.

Let (0, ¢) be a domain spread over the complex projective space P(E) induced from E. E- {0} is the total space of the holomorphic principal bundle over P(E) with the complex multiplicative group C*. We consider the fibre product X of 0 and E- {0} given by

(2.5) X= {(^z, ^w) ^{En X} (E- {0} ); cf;(^z) ⁼ Q(^w)}.

We denote by

¢

^and

Q

projections of the fibre product

X

^into

E- {0}

^{and into n}

respectively. Then

(X,'¢)

is a domain spread over

E.

For any

(z, w) EX

and for any

A E

^{C*, we set}

(2.6) A· (z,w) = (z, Aw)

Then points

A· (z,w)

^{of S1 x}

(E- {0})

belong to

X

^{for all}

A E

C*. The mapping

(A'

^X

)

^__..

A .

^X is a holomorphic mapping of C* ^X

X

^onto

X.

^{Then n} is the quotient

space of X by this C* -action and

Q

is the quotient map of

X

^{onto n.}

X

^{is the}

total space of a holomorphic principal bundle over n with the complex multiplicative group C*. We have the following commutative diagram:

X Q (2,7)

E- {0} --Q- P (E)

Let f be a holomorphic function in

X.

^{We set}

(2.8) ](x) = (1/27r) [� ^j(e'9

^•

^x)de

for every

x E X.

^Then

]

i a holomorphic function in X and we have

f ( e

^{i7J • X}

) = ] (

^X

)

for every 17

E [0, 21r)

and for every

x E X.

By the identity theorem of a complex variable holomorphic function theory, we have

(2.10) ](A·x)=](x)

7

- - -1

for every ,\ E

C*.

Therefore f is constant on Q ( ^z

)

for every ^z E

0.

We define a holomorphic function

!*

in n by

(2.11)

j*

( ^z

)

= f ^-( ^--Q ¹ ( ^z

))

for every ^z E

0.

We have

(2.12)

(g

0

Q)*

=

g

for every

g

E H (

0).

Hence we obtain the following lemma.

LEMMA 2.1. For any f E

H(X)

1 a holomorphic function] in

X

defined by (2.8)

-

^-1

is constant on Q

(

z

)

for every z E

0.

Thus we can define a holomorphic function

!*

inn by (2.11).

Let

F

be a closed linear subspace of E. We denote by XF and by OF regions spread over F and spread over the complex projective space

P(F)

induced from F,

respectively, defined by

(2.13)

(2.14)

XF is a holomorphic principal bundle over Op with the complex multiplicative group

C*.

We have the following commutative diagram induced from the commutative diagram (2.7):

(2.15)

F-

{0} -

^-Q-1

(-F

-_

- 0 { - } )

^__... p

(F)

Let (D, ¢) be a region spread over a complex projective space P(E) induced from a complex Banach space E. Then the region (D, ¢) is said to be pseudoconvex if, for every f^E E^'- {0} and for the open set U(f), defined by (2.3), of P(E), the open set ¢-1(U(f)) of D satisfies the Kontinuitatssatz.

LEMMA 2.2. Let E be a complex Banach space and (D, ¢) be a domain spread over the complex projective space P(E). Suppose that D is not homeomorphic to P(E) through ¢. Then for any finite dimensional linear subspace F of E and for any connected component VF of DF 1 there exist a finite dimensional linear subspace

G of E and a connected component Ve of De satisfying the following conditions:

(1) VF is a closed complex submanifold of Ve.

(2) Ve is not homeomorphic to P( G) through ¢1Ve .

PROOF. By Remark 1.1 and by the commutative diagram (2.15) there exist a finite dimen ional linear ubspace F0 of E and a connected component VFo of DFo

such that VFo i not homeomorphic to P(F^o) through c/JIVFo. We take a point z of VF

and a point w of VFo· By Remark 1.1 and by the commutative diagram (2.15), there exists a finite dimen ional subspace F1 such that a connected component VF1 of DF1

contains the et { z, w} . Let G be the complex vector space spanned by all elements of the union F U F0 U F1. Then P(F) and P(F0) are closed complex submanifolds of

P( G). We denote by Ve the connected component of De containing the set { z w }.

Since (Ve, ¢1Ve) is a domain spread over P( G), both VF and VFo are closed con1plex submanifolds of Ve. Then Ve satisfies the required condition (1) and (2). This complete the proof.

LEMMA 2.3. Suppose that D is not homeomorphic to P(E) through ¢ and that D is pseudoconvex. Then1 for any finite dimensional linear subspace F of E1 DF is

a Stein manifold. ^X satisfies the ^J( ontinuitiitssatz.

9

PROOF. Let F be a finite dimensional linear subspace of E. Let

Vp

be any component of

Op.

By Lemma 2.2 there exists a finite dimensional subspace

G

of E and a component

Vc

of

00

satisfying the conditions

(1)

and

(

2

)

in Lemma 2.2.

Since

n

is p eudoconvex,

V0

is also pseudoconvex. By Fujita

[

4

]

, Kiselman

[

9

]

and Takeuchi

[

22

]

, the pseudoconvex domain

V0

spread over P(

G)

is a Stein manifold.

Since

Vp

is a closed complex submanifold of the Stein manifold

V0, Vp

is a Stein Inanifold. Thus

nF

is a Stein manifold.

XF

is the total space of a holomorphic principal bundle over the Stein manifold

nF

with the complex multiplicative group C*. Therefore

XF

is a Stein manifold by Matsushima and Morimoto

[

12

]

. Since

(X,'¢)

is a domain spread over E,

X

satisfies the Kontinuitatssatz by Noverraz

[

17

J

. This co1nplete the proof.

LEMMA 2.4 With the assumption of Lemma 2.2 the following conditions are

equivalent:

( 1) n

is pseudoconvex.

(

2

) nF

is a Stein manifold for every finite dimensional linear subspace F of ^{E .}

PROOF. It follows from Lemma 2.3 that

(1)

implies

(

2).

We will how that

(

2

)

implies

(1).

Let

f

be an element of E'-

{0}.

^BY the assumption for every finite dimensional linear subspace F of E with dime F 2: 2 and F ct.

{! = 0}, nF

is a Stein manifold. We et H

= ¢-1 ( {[�]

^E P(F);

f(�) = 0} ).

Since His a hypersurface of

Op

and

Op

ⁿ

¢-1(U(f)) = Op\H, Op

ⁿ

¢-1(U(f))

is a Stein manifold.

¢-1 ( U (f))

and

nF

ⁿ

¢-1 ( U (f))

are identified with regions spread over the Banach space

{f = 0}

and spread over the finite din1ensional subspace

{f = 0}

ⁿ F of

{f = 0}

respectively. Therefore by Noverraz

[

17

]

the domain

¢-1(U(f))

satisfies the Kontinuitatssatz. Thus

n

is pseudoconvex. This completes the proof.

3 Some properties of the fibre product X.

In this section we will research some properties of the fibre product X, defined in the preceding section, of

n

and

E-

{0} for complex Banach space

E

with a Schauder

basis and for a pseudoconvex domain

(n,

¢)spread over the complex projective space

P(E).

Let

E

be a complex Banach space with the norm

II

·

II

and a Schauder basis

{ej}1=1. Let (0, ¢)be a pseudoconvex domain, which is not homeomorphic to

P(E)

through ¢, pread over the complex projective space

P( E).

Since

n

is pseudoconvex, by Lemma 2.3 X satisfies the Kontinuitatssatz. By Noverraz[17], we have the following Lemma 3.1.

LEMMA 3.1. -log dx is a continuous plurisubharmonic function in ^X where ^dx

is the boundary distance function on ^X. For any finite dimensional linear subspace

F

_of

E,

¢ --1

(F)

is a Stein manifold.

We can choose a Schauder basis { e1 }f=1 ^of

E

such that the intersection of the image of

'¢

and the linear pace { Ae1; ^{). E} C} is nonempty. For every � ^E

E

� ^can

be represented in a unique way

00

(3.1)

We denote by

En

the linea span of the set { e1, e2, ^{• • • ,} en}, ^{and by Un} the mapping

of

E

onto

En

defined by

n

(3.2) _u

n

(�^{) =}

L

_�JeJ"

j=l

We denote by

J.-ln

a continuous linear functional of

E

defined by (3.3)

11

LEMMA 3 .2.

There exist a norm 111·111 of

^E

and positive constants

^c1

and

^c2

satisfying the following conditions:

(1)

c1

II � 11�111�111�

^c2

II �II

(2) lllun(

�)Ill� Ill� Il

l

for every�

^{E E.}

for every positive integer

^n.

The proof of Lem1na 3.2 is in Singer[21]. The condition

(1)

of Lemma 3.2 implies that the Banach space

(E, 111·111)

with the norm

111·111

is equivalent to the Banach space ^E with the original norm

II · II·

Therefore we may assume that the norm ^E satisfies the condition

(3.4)

II

un(�)

11�11 �II

for every positive integer ^n.

Let

x0

be a point of ^X with ¢(

x0)

^{E E1.} We may assun1e that the norm

II · II

^of

E is chosen such that dx (

x0) � 1.

For every ⁿ we set

(3.5)

(3.6) An= {x ^{E X;} sup

II

^{Urn o}

¢(x)- ¢(x) II<

dx(x)}, rn;?:n

(3.7)

for every x ^E An. Then suprn;?:n

II

^{Urn o}

¢(

x)

- ¢( x) II

is continuous on ^X, and An is a.n open subset of X. vn is a holomorphic mapping of An into Xn for every n.

Let (Y, 1j;) be a. region spread over a. complex Banach space ^F. Then we use the notation dy(A) = inf{dy(x); x ^E A} where A is a. subset of ^Y.

The proof of the following lemma. is in Lemma. 54.5 of Mujica.[l3].

LEMMA 3.3.

There exist two increasing sequences { Bn}�=l and { en}�=l of open sets Bn and en of X such that

(a)

Xo

c en c Bn cAn for every n 2 1, X= u�=lBn = u�=len.

(b) dAn (Bn) 2 2-n and Bm n Xn is relatively compact in Am n Xn for every m, n 21. (c) dcm+1

(

em) 22-m-land vn(em) C BmnXn for every m 21 and

every n 2m.

For every x EX, we define the sets V(x) and S(x) by

(3.8) v (X)

= {A .

X;

A E C*}'

(3.9)

S

(X

) = { e iB ^•

X

;

0

_::; f) ::; 21f} .

Let ]{ be a compact subset of a Stein manifold S. We use the notation

(

3

.1

0

) 1<(5) ={xES· if(x)i::; sup if(y)i for all f E H(5)}.

yEK

The set I<(S) is the holomorphically convex hull of]{ in the Stein manifold ^S. If

1<(5)

= I<, I< is said to be Runge in S. Let ⁵¹ be a Stein manifold and

52

_{be a}

Stein open u bset of

5

1

. 52

i said to be Runge relative to S1 if, for any compact subset ]{ of

52,

I<(S1) is a compact subset in

52.

We denote by ]{ n the holomorphically convex hull of the topological closure

of the set Bn n Xn+l in the Stein manifold Xn+l· Since Xn+l is a Stein mani­

fold, I<n is a compact subset of Xn+l and Runge in Xn+l· On the other hand supm�n II Um o �(x)- �(x) II is continuous in X, and supm�n log II Um o

'¢-

�(x) II -log dx(x) is a continuous plurisubharmonic function of ^�Y into

[

-oo, oo

)

. There- fore by N arasimhan[15], Ann Xn+l is Runge relative to Xn+l and ]{n is compact in

13

LEMMA 3.4. Let { cn}�=1 be a sequence of points of X such that Cn _E Xn,

Cn

t/:.

Xn-1 and V(cn) C X\I<n· Then, for any sequence {>-n}�=1 of positive numbers,

there exists a sequence {fn}�=1 of holomorphic functions fn in Xn such that (3.11)

(3.12)

for any

x

_E f{n, and (3.13)

for any

x

_E

S(

en) where Refn represents the real part of fn.

PROOF. We will how this lemma by induction with respect to n. We set

/1(x)

=

_>-1 for every

x

_{E X. Then f1 atisfies (3.13).} We assume that there exist holomorphic functions fk in Xk (1 � k � ⁿ

)

with (3.11), (3.12) and (3.13). We set

(3.14)

g(x) =

fn o vn(x

)

for every

x

_E Xn+1 nAn· Closed subsets I<n _U Xn and (Xn+1 \An) are mutually disjoint because ]{ n is a compact subset of Xn+l n An- Therefore there exist a C00-function

r;

_in Xn+1 such that

r; =

1 on a neighborhood of I<n _U Xn, and that

r; =

₀ on a neighborhood of (Xn+1 \An)·

We consider a 8-equation on Xn+1:

(3.15)

where f-lJ are defined in (3.3). Since Xn+1 is a Stein manifold, and since the right hand side of (3.15) is 8-closed, there exists a C00-function v on Xn+1 satisfying

(3.15). We set

(3.16) h

(

X

) = rJ (X) g (

X

)

- ( f.-ln + 1 0

¢ (

X

))

V

(

X )

for every x E Xn+l· Then h is holomorphic in Xn+l and satisfies hiXn ⁼ fn· Since

v is holomorphic in a neighborhood of a Runge compact subset ]{ n of Xn+l, by Oka-Weil theorem there exists a holomorphic function w in Xn+l such that

(3.17) jv(x)- w(x)l < 1/(2n+l M)

for every x E I<n where M ⁼ sup{IJ.-ln+l ^o �(x)j; x E I<n}· We set (3.18) F(x) ⁼ h(x) + (J.-ln+l ^o �(x))w(x)

for every x E Xn+l· Then we have

(3.19)

for every x E ^]{ n·

We set

(3.20)

(3.21)

We denote by T the holomorphically convex hull ofT in Xn+l· Since Xn+l is Stein, Tis compact in Xn+l·

We will show that T ^c vn+l ^u ]{n- Let ^X be a point of Xn+l \(Vn+l ^u I<n)·

Since Xn+l is a Stein manifold, by Oka-Cartan theorem there exists a holomorphic function ^s in X n+l with ^{s =} 0 on vn+l and with ^s( X) ⁼ 1. Since ]{ n is a Runge compact sub et of Xn+l, there exists a holomorphic function t in Xn+l, such that

lt(x)l > 1 and II t IIKn< 1/(11 ^s IIKn +1) where II ^s III(n and II t IIKn represent supremums of functions j^s(·)l and j t(·)l, respectively, on the compact set I<n- Then

we have ls(x)t(x)l > 1 and sup{js(y)t(y)j^; yET}< 1. Therefore x cannot belong

� �

to T. Thus we have T ^c Vn+l ^U I<n-

15

Since T is a Runge compact subset of Xn+b there exist Stein neighborhoods

D11 and �2 of ('"F ⁿ Vn+1) and of I<n, respectively, in Xn+1 with �1 n �2 = 0. We set L = sup{IF(x)l; x E S(cn+1)}. We define a holomorphic function a in a Stein manifold �1 n vn+1 by

(3.22)

for every ).. . Cn+1 E �1 n Vn+1 (>.. ^{E c-}{0} )^{. Since �1 n} vn+1 is a closed complex submanifold of �1, by Oka-Cartan theorem there exists a holomorphic function A

in �1 such that AIVn+1 n �1 = a. We define a holomorphic function Bon �1 U �2 by Bl�1 =A and Bl�2 = 0. Since �1 U�2 is a neighborhood of the Runge compact subset T in Xn+1 there exists a holomorphic function G on Xn+1 such that

(3.23) IG(x)- B(x)l < 1/{2n+1(L' + 1)}

for every x E T ^where L' = sup{IMn+1 o¢(x)l; x E S(cn+1)UI<n)}. We set fn+1(x) =

F(x) + (Mn+1 o ¢(x))G(x) for every x E Xn+1· By (3.19) and (3.23) we have

(3.24)

for every x E I<n- By (3.22) and (3.23) we have

(3.25)

for every e E R. Since fn+11Xn ⁼ fn, this completes the proof.

LEMMA 3.5. Let { En}�=1 be a sequence of positive numbers with 2:::�=1 En < ^oo

and {fn}�=1 be a sequence of holomorphic functions fn in Xn such that fn+11Xn ⁼ fn and lfn+1(x)- fn o vn(x)l <En for every x E I<n· Then there exists a holomorphic function f in X such that fiXn = fn·

PROOF. Since, by Lemma 3.3, Vn+J(Cn+J-l) ^c Bn+j-l ⁿ Xn+j ^c I<n+J-l and

Cn ^c Cn+J-l, we have lfn+jOVn+J(x)- fn+J-lOVn+J-l(x)l ⁼ lfn+j(Vn+J(x))- fn+J-lo Vn+j-l(vn+J(x))l < En+j-l for any positive integers nand j and for any x ^E Cw Thus for any m, n we have

m

]=l

m 00

<

L

tn+J-l :S

L

EJ

J=l J=l

for every x ^E Cn. Therefore the sequence {fn o vn}�=l converges uniformly on each

Cn to a function f ^E H(X). Then f satisfies fiXn = fn· This completes the proof.

We can obtain the following two lemmas by the application of Lemma 3.4 and Lemma 3. 5.

LEMMA 3.6. With the conditions of Lemma ^3.4, there exists a holomorphic function f in X such that Re f ( x) 2:: An for every n and for every x ^E

S(

Cn) .

LEMMA 3. 7. Let ^F be any finite dimensional complex linear subspace of E.

Then the restriction mapping of H (X) into H

( �

-l

(F))

is surjective.

4 Proof of Theorem 1 and Theorem 2.

In order to prove Theorem 1 and Theorem 2, we will prepare some lemmas.

Throughout this section E means a complex Banach space with a Schauder basis

{en}�=l and

(0,</J)

means a domain, which is not homeomorphic to the projective space P(E) _through

</J,

spread over P(E).

LEMMA 4.1 . If

0

is a domain of holomorphy,

0

is pseudoconvex.

PROOF. For any continuous linear functional f of E and the open set

U(f)

⁼

{[�] ^E P(E); f(�)

-=/

0}, we have only to show that the domain

</J-1(U(f))

satisfies 1 7

the Kontinuitatssatz. Since there exists a biholomorphic mapping p of U (f) onto the complex Banach space L = {� ^{E E;} J(�) ⁼ 0}, the domain (¢-1(U(J)),p ^o (¢j-1(U(f))))

is a domain spread over L. Since 0 is a domain of holomorphy and since, for

any sequence {xn}�=l of q;-1(U(f)) converging to a point of 0\¢-1(U(f)), the set

{p ^o ¢(xn)} is an unbounded subset of L, q;-1(U(J)) is also a domain of holomor­

phy. By N overraz[17], q;-1 ( U (f)) satisfies the Kontinuitatssatz. This completes the proof.

W ith the conditions and notations in Section 3, we set

( 4.1)

___ ....__

for each n. 5( J( n) ^{is COin pact in} Xn+1 nAn. We denote by 5( f(n) the holomorphically

---

convex hull of 5(I<.:n) in .Xn+l· Since Ann Xn+l is Runge relative to .Xn+l, 5(1(n) is a compact subset of Ann Xn+l· We set eie^. Cn ⁼ { eie^{. X ; X E} Cn}· For any e E R, we have

(4.2) ( 4.3)

. �

e18 ^• Cn n Xn+l ^C 5(J(n) ^C 5(J(n),

---

vn(eie ^· Cn) ^C 5(J(n) ^C 5(J(n)·

Hereafter we assun1e that 0 is pseudoconvex in a series of lem1nas.

LEMMA 4.2. Then for any holomorphic function f in Xn there exists a sequence

{fn+k}k=O of holomorphic functions in Xn+k satisfying the following conditions:

(1) fn ⁼ J,

(2) fn+k 1./Yn+k-1 ⁼ fn+k-1 ¹

..._

(3) lfn+k(x)- fn+k-1 ^o Vn+k-1(x)l ^< 1/2n+k for every ^{X E} 5(f(n)·

PROOF. We can prove this lemma by the same way as the proof of Lem1na 3.4.

....

REMARK 4.3. By the same way as the proof of Lemma 3.5, we can prove that there exists a holomorphic function F in ^X such that FIXn+k = fn+k and F(x) =

limk-+oo fn+k o Vn+k(x) for every x ^{E X.} By ( 4.2) and ( 4.3), we have

IF(x)l m-+oo lim lfm o Vm(x)l

< li�_;;.!:,P

Lt

lfk o vk(x) - fk-1 o llk-J (x )I ⁺ I!N o vN(x )I

}

< 2-N ⁺ sup{I!N o vN(Y)i; y ^E S(CN )} ^<(X)

for every ^N � ⁿ and for every x ^E S(CN) where S(CN) is the set {eie ^{· z;} (B, ^z) ^E

R ^x CN}· Thus we have sup{IF(x)l; x ^E S(CN)} ^<(X) for every ^N � ^1.

We denote by Dm an open sub et of n defined by Dm = Q( Cm) for every ^m � 1.

LEMMA 4.4. For any holomorphic function f in c/J-1(P(En)) there exists a

holomorphic function F inn such that FI¢Y-1(P(En)) ⁼ f and sup{IF(x)l ; x ^E Dm} ^{< (X)} for every ^m � ^1.

PROOF. We consider a holon1orphic function gin Xn defined by g = f o ( QIXn)·

By Lemma 4.2 and by Remark 4.3, there exist a holomorphic function Gin X such that GIXn ⁼ g and sup{IG(x)l ; x ^E S(Cm)} ^<(X) for every ^m � ^1. We set

{21f G(x) = (1/27r)

Jo ^{G(eie · x)dB}

for every x ^E X. Then G is a holomorphic function in X and constant on Q ^-1 ( ^z) - --1 for every ^{z E D.} We define a hol01norphic function F by F(x) = Go Q (^z) ^for every ^{z E} n. Then we have FI¢Y-1(P(En)) = f and sup{IF(x)l ^{; z E} Dm} � sup{IG(x)l ^{; z E} S(Cm)} ^<(X) for every ^m � ^1. This con1pletes the proof.

19

LEMMA 4.5. For any different points z and w in n, there exists a holomorphic function f in n such that f(z) i= f(w) and that sup{lf(p)l ; p E Dm} < ^OC> for

every m 2 1.

PROOF. There exist two diffent points x and y in X such that Q(x) ⁼ z and

Q(y) ⁼ w. There exists a positive integer N such that the set {x, y, vN(x), vN(y)}

is contained in CN and that Q(vN(x)) i= Q(vN(y)). Then the compact sets S(x), S(y), S(vN(x)) and S(vN(y)), defined in (3.9), are contained in S(CN)· We consider closed ubmanifolds V(vN(x)) and V(vN(Y)), defined in (3.8), of the Stein manifold

XN. By Oka-Cartan theorem, there exist a holomophic function g in XN satistying

giV(vN(x)) ⁼ 2 and giV(vN(y)) ⁼ 0. By Lemma 4.2, there exists a sequence

{gm}�=N of holomorphic functions 9m in XN+m such that 9miXm-1 ⁼ 9m-1, 9N ⁼ g

and 19m o vm(t)- 9m-1 o Vm-1(t))i < 1/2m for every m > N and every t E S(Cm-1)·

Let G be a holomorphic function defined by G(t) ⁼ limm-+oo 9m o vm(t) for every

t EX. Then we have IG(t) ^- go vN(t)i � 1/2N for every t E S(CN)· Thus we have Re G(eie ^· x) 2 Re g o vN(eie ^· x)- 1/2n 2 3/2 andRe G(eie ^· y) :::; Reg o

vN(eie ^· y) + 1/2N :::; 1/2. By Remark 4.3, the holomorphic function Gin X satisfie

sup{IG(t)l ; t E S(Cm)} < OC> for every m 2 1. We set

for every t E X. Then G is a holomorphic function in X and constant on Q -1 ( () - --1

for every ( E n. We set /(() ⁼ Go Q (() for every ( E n. Then f is a holo-

morphic function and satisfies Re f(w):::; 1/2 < 3/2:::; Re f(z). Moreover we have

sup{lf(()i ; ( E Dm}:::; sup{IG(t)i ; t E S(Cm)} < OC>^. f satisfies the requirement of this lemma. This completes the proof.

We set 1) ⁼ {Dn}�=1 and set lfln ⁼ sup{lf(x)l ; x E Dn} for every f E H(n)

...

and every n � 1. We denote by A('D) the Frechet space defined by

A('D) ⁼ {f ^E H(rl) ; lfln < oo for every n}.

LEMMA 4.6. For each countable set P of rl there exists a function g ^E A('D)

such that g( x) f. g(y) for all ( x, y) ^E P x P\6. where 6. is the diagonal set of the product P x P.

PROOF. By Lemma 4.5, the set Sxy ⁼ {g ^E A('D) ^; g(x) f. g(y)} is nonempty

for each (x, y) ^E P x P\6.. The set Sxy is open in A('D). We claim that Sxy is dense in A('D). Let f be an element of A('D) with f � Sxy· We choose g ^E Sxy and set

9n ⁼ f ⁺ (1/n)g. Then we have 9n ^E Sxy for every n and 9n ^� f in A('D). Since

A('D) is a Baire space, the set S ⁼ n{Sxy ^; (x, y) ^E P x P\6.} is dense in A('D),

and in particular nonempty. Thi completes the proof.

PROOF OF THEOREM 1. It follows from Lemma 2.4 that (1) and (2) are

equivalent. It follows from Lemma 4.1 that (3) implies ( 1). It is clear that ( 4)

implies (3).

Now we will show that ( 1) implies ( 4). Let En be the linear span of the set

{e1, ^{· · ·},en}^· We may assume that Q(e1) ^E ¢(0). Since P(E) is separable, there exists a countable dense subset D of P(E). We set P = ¢-1(D). Then P is a countable dense subset of fl. By Lemma 4.6 there exists a holomorphic function

g ^E A('D) such that g(x) f. g(y) for every (x, y) ^E P x P\6.. Let d be the distance of

P(E) defined by (2.4). We denote by rln the region, defined by rln ⁼ ¢-1(P(En)),

spread over P(En) for every n. We denote by dn the boundary distance function of the region (rln, ¢1rln) with respect to diP(En)· For each X E nn we denote by

Bn( x) the open neighborhood, which is homeomorphically mapped by ¢1rln onto the set {( ^E P(En) ; d(¢(x)^, () ::; dn(x)}, of x in rln. We set Ln ⁼ Q(I<n) for

21