SOME CHARACTERIZATIONS OF THE PROPERTIES (DN) AND … † e

SOME CHARACTERIZATIONS OF THE PROPERTIES (DN) AND … † e

LE MAU HAI and NGUYEN VAN KHUE

Abstract

The aim of this paper is to show that

Hw…B;F† ˆH…B;F†

for every LB¹- regular compact setBin a Frechet spaceEif and only ifFis a Frechet space having property (DN). At the same time, the equivalence between the existence of a LB¹- regular compact setBin a Schwartz - Frechet spaceEwith an absolute basis and the property …†e ofEis also established here.

1. Introduction

LetEbe a Frechet space with the topology defined by an increasing system of semi-norms

k k_k . For each subset B of E we define

B : E! ‰0;‡ 1Šgiven by u

Bˆsupn

ju…x†j:x2Bo whereu2E, the topological dual space ofE.

Instead of _Uq we write _q, where Uq ˆn

x2E:kxk_q 1o Now we say thatE has the property

(DN) 9p 9d >0 8q 9k;C>0: ^1‡d_q C _k ^d_p …† 8pe 9q;d >08k9C>0: ^1‡d_q C _k ^d_p (LB¹) 8

n n2N" 1 8p 9q8n0 9N0n0; C>08u2E 9n2N:n₀nN₀ u^1‡n

q C _{n p} ⁿ

The above properties have been introduced and investigated by Vogt (see [9], [10]).

Received March 25, 1998.

Note that the following equivalent form of the property (DN) has been formulated by Zahariuta in [12]

…(DN)†_Z 8p 8q;d >0 9k;C>0: ^1‡d_q Ck k_kk k^d_p

Let E and F be locally convex spaces and let E be open, 6ˆ ;.

f :!F is called Gaªteaux - holomorphic if for everyy2F, the topologi- cal dual space of F, the function yf :! C is holomorphic. This means that its restriction to each finite dimensional section ofis holomorphic as a function of several complex variables.

A function f :!F is called holomorphic if f is continuous and Gaªteaux - holomorphic on.

Now letBbe a compact subset in a locally convex spaceEandF a locally convex space. By the standard notationH…B;F†denotes the space of germs of holomorphic functions onBwith values in F with the inductive limit to- pology.

Recall thatf 2H…B;F†if there exists a neighbourhoodV ofBinE and a holomorphic function^f :V !F whose germ onBisf. AF-valued function f on B is called weakly holomorphic onB if for every x2F, the topolo- gical dual space ofF equipped with the strong topology…F;F†,xf can be extended holomorphically to a neighbourhood ofB. ByHw…B;F†we denote the space ofF-valued weakly holomorphic functions onB.

For details concerning holomorphic functions and germs of holomorphic functions on compact subsets of a locally convex space we refer to the books of Dineen [1] and Noverraz [6].

One of aims of this paper is to find some necessary and sufficient condi- tions for which

Hw…B;F† ˆH…B;F† …!†

The statement…!†has been investigated by several authors. Siciak in [8] and Waelbroeck in [11] have considered this problem for the case, where dimE<1andF is a Baire space. After that, in [4] N. V. Khue and B. D.

Tac have shown that…!†holds in the case, whereF is still Baire and either Eis a nuclear metric space orF is nuclear. The Baireness ofFplays a very important part in the works of the above authors. However, at present, when F is not Baire, in particular,F is a Frechet space which is not Banach…!†

has not been established by any authors.

In the second part of this paper we give a characterization of the property (DN) by showing that…!†holds ifF is a Frechet space having the property (DN) and Bis a LB¹ - regular compact set in a Frechet space E, where a compact setBin a Frechet spaceEis said to be LB¹- regular if‰H…B†Šhas

property …LB¹†. Next, from the obtained result of the second section the third section is devoted to establishing some characterizations of the prop- erty…†e of a Schwartz - Frechet spaceE with an absolute basis.

In through paperF_bor denotes the spaceFequipped with the bornological topology associated with the topology ofF. This is the most strong locally convex topology on F having the same bounded subsets as the …F;F† - topology.‰F_bor Š is equipped with the…F;F†- topology.

Acknowledgement. The authors would like to express many thanks to the referees for their helpful remarks and suggestions.

2. Characterization of (DN)

The main result of this section is the following 2.1. Theorem.Let F be a Frechet space. Then

Hw…B;F† ˆH…B;F† …!†

holds for every LB¹- regular compact set B in a Frechet space E if and only if F has property(DN).

In order to prove Theorem 2.1 we need some lemmas.

2.2. Lemma.Every LB¹- regular compact set B in a Frechet space E is a set of uniqueness,i.e.if f 2H…B†and f

Bˆ0then f ˆ0on a neighbourhood of B in E.

Proof. Let fV_ng be a decreasing neighborhood basis of B in E. Given f 2H…B†with f

Bˆ0, choosep1 such thatf 2H¹…Vp†. For eachnp, put

"nˆ kfk_nˆsupn

jf…z†j:z2Vn

o Thenf"_ng #0. By the hypothesis

H…B†

has property (LB¹) and employ- ing this withfng ˆn 

log 1

"_n r o

" 1we have

9q 8n0 9N0n0; Cn0>08m>0 9km:n0 kmN0 : f^m1‡km

q Cn0f^m

kmf^mkm

p

which yields

f^1‡km

q Cn^m10f

kmf^km

p

Choosen0kN0 such that

Cardn

m:k_mˆko

ˆ 1 Then

kfk_qf^1‡1k

k f^1‡kk

p

ÿ

"k_1‡¹

kÿ

"p_1‡^k

k !0

ask! ‡1.

Hencef

Vq ˆ0. Lemma 2.2 is proved.

2.3. Lemma. Let F be a Frechet space having property(DN). Then F_bor has property(DN).

Proof. LetfUng be a decreasing neighbourhood basis of 02F. SinceF has property (DN) we have

9p8q 9k; C>0:k k_qrk k_p‡C rk k_k for allr>0, or in equivalent form [10]

9p 8q 9k; C>0:U_q⁰rU_p⁰‡C

rU_k⁰ for allr>0 Foru2

F_bor

andr>0 we have u

q ˆsupn

ju…x†j:x2U_q⁰o

supn

ju…x†j:x2rU_p⁰‡C rU_k⁰o rsupn

ju…x†j:x2U_p⁰o

‡C rsupn

ju…x†j:x2U_k⁰o

ˆ

ˆru

p ‡C ru

k

Hence F_bor

has property (DN).

Lemma 2.3 is proved.

Proof of Theorem 2.1 Sufficiency. It suffices to prove that Hw…B;F† H…B;F†. Letf 2Hw…B;F†andF has property (DN), whereBis a LB¹ - regular compact set in a Frechet spaceE. By Lemma 2.2B is a set of uniqueness and, hence, we can consider the linear map ^f :F_bor !H…B†

given by

^f…x† ˆxcf

for x 2F_bor , where xcf is a holomorphic extension of xf to some neigh- bourhood ofBin E. Still by the uniqueness ofBit follows that^f has closed graph. On the other hand,F_bor is an inductive limit of Banach spaces,H…B†

is an (LF) - space so by closed graph theorem of Grothendieck [3]^f is con- tinuous. Since^f maps bounded subsets of F_bor to bounded subsets of H…B†

then the dual mapf^⁰:‰H…B†Š! ‰F_bor Š is also continuous. By the hypoth- esis ‰H…B†Š has property (LB¹) and by Lemma 2.3

F_bor

has property (DN). From a result of Vogt [9] it follows that there exists a bounded subset LH…B†such that^f⁰…L⁰†is a bounded subset of ‰F_bor Š, whereL⁰ denotes the polar ofLin‰H…B†Š. Henceÿ^f⁰…L⁰†₀

is a neighbourhood of 02

…F_bor †

. PutW ˆÿ^f⁰…L⁰†₀

\F_bor . ThenW is a neibourhood of 02F_bor . We have

^f…W† L⁰⁰\H…B†

whereL⁰⁰is the bi-polar ofL. HoweverL⁰⁰\H…B†is the closure of the ab- solutely convex envelope of Land, hence, it is a bounded subset of H…B†.

This shows that ^f…W† is bounded inH…B†. By the regularity of H…B†there exists a neighbourhoodU ofBin E such that^f…W†is contained and boun- ded inH¹…U†, the Banach space of bounded holomorphic functions on U.

From the absorption of W it follows that ^fÿ F_bor

H¹…U†. Now we can define a holomorphic function

g:Uÿ!

F_bor given by

g…z†…x† ˆ^f…x†…z†

forz2U,x 2F_bor .

We see that g…z†…x† ˆ^f…x†…z† ˆf…z†…x† for every z2B, x 2F. This yieldsg

Bˆf and sinceBis a set of uniqueness,g…U† F.

Necessity.By Vogt [9] it suffices to show that every continuous linear map T from H…† to F is bounded on a neighbourhood of 02H…†, where ˆ fz2C:jzj<1g. Consider T:F! ‰H…†ŠH…†. Since T…x† 2 H…†for allx2F, we can define a mapf :! ‰FŠgiven by

f…z†…x† ˆz…T…x††

forx2F,z2, where _zis the Dirac functional defined byz _z…'† ˆ'…z† for '2H…†:

From the weak continuity ofT andzwe infer thatf…z†is…F;F†-con- tinuous and, hence, f…z† 2F. Moreover, f 2H_w…;F†. Since is LB¹- regular it follows thatf 2H…;F†. Thus there exists a neighbourhoodV of such thatf 2H¹…V;F†. Hence,Bˆf…V†is bounded in F. It is easy to see thatTis bounded onB⁰. PutCˆT…B⁰† ‰H…†Š andUˆC⁰. Then Uis a neighbourhood of 02H…†andT…U† B⁰⁰is bounded inF.

Theorem 2.1 is proved.

3. Some characterizations of…†e

This section is devoted to give some characterizations of the property…†e on a Schwartz - Frechet spaceEwith an absolute basis.

The following theorem is the main result of this section.

3.1. Theorem. Let E be a Schwartz - Frechet space with an absolute basis.

Then the following are equivalent

(i) There exists a compact set B of uniqueness in E such that H_w…B;F† ˆH…B;F†for all Frechet spaces F having property(DN).

(ii) There exists a compact set B in E such that‰H…B†Šhas property(LB¹).

(iii) There exists a compact set B in E which is not polar.

(iv) E has the property…†.e Proof.

(ii))(i) by Theorem 2.1.

Now we give the proof (i)) (iii). The implication (i)) (iii) is obtained from the following proposition

3.2. PropositionLet B be a compact set of uniqueness in a Frechet space E having a Schauder basis and let

Hw…B;F† ˆH…B;F† for every Frechet space F 2(DN). Then B is not polar.

Proof. Otherwise, assume that B is polar. Choose a plurisubharmonic function'onE such that'6ˆ ÿ1and

'

Bˆ ÿ1

Consider the Hartogs domain'given by 'ˆn

…z; † 2EC:jj<e^ÿ'…z†o

Since'is plurisubharmonic,_'is pseudoconvex. BecauseE has a Schauder basis so_' is the domain of a holomorphic function f. Write the Hartogs expansion off

f…z; † ˆX¹

nˆ0

h_n…z†ⁿ where

h_n…z† ˆ 1 2i

Z

jjˆe^ÿ'…z†ÿ

f…z; †

^n‡1 d; for >0:

By the upper semi-continuity of'it follows that h_n is holomorphic on E for alln0. Consider the functiong:B!H…C†, given byg…z†…† ˆf…z; †.

Let 2 ‰H…C†Š be arbitrary. There exists r>0 such that 2 ‰H…r†Š, whereˆ

2C:jj 1 . From the openness of' it follows that there exists a neighbourhood V of B such that Vr'. By the absolute convergence of the series P¹

nˆ0hn…z†ⁿ on Vr it follows that g2H…V†

and, hence, g2H_w…B;H…C††. Applying the hypothesis to F ˆH…C†which has property (DN) we find a neighbourhood U of B in E and a bounded holomorphic function^g2H…U;H…C††which is a holomorphic extension of g. We can write

^

g…z; † ˆX^‡1

nˆ0

^ g_n…z†ⁿ

where^g_n…z†is holomorphic onU for alln0. Choose a neighbourhoodW of B such that W U and W2_'. Define two holomorphic func- tions

H:W ! H¹…†

z 7! …h₀…z†;h₁…z†;. . .†

G:W ! H¹…†

z 7! …^g₀…z†;^g₁…z†;. . .†

SinceH¹…†is a Banach space andH

BˆG

B, it follows that there exists a neighbourhoodW₁ of B in W such that bg

W1ˆf

W1. Let X be a con- nected component of W1. Since XC is connected, bg

Xˆf

X,

X' and' is the domain of existence of f we have XC'. Hence'

Xˆ ÿ1. This is impossible.

Proposition 3.2 is proved.

The following proposition gives the implication (iii))(iv).

3.3. Proposition. Let E be a Frechet space. If there exists a non polar compact set in E then E has property…†.e

Proof. By a result of Dineen - Meise - Vogt [2, Corollary 8 and Theorem 10].

Finally, the implication (iv))(ii) is given by the following proposition.

3.4. Proposition.Let E be a Schwartz - Frechet space with an absolute ba-

sis. If E has the property …†e then there exists a balanced convex compact subset B of E such that‰H…B†Š has property(LB¹).

Proof. Let fe_jg_j1 be an absolute basis for E. From the hypothesis, by Vogt [9], there exists a balanced convex compact setB₁ inEsuch that

…eB1† 8p9q; d>0; C>0 : k k^1‡d_q Ck k_B1 k k^d_p …1†

On the other hand, sincefejg_j1 is an absolute basis it follows thatke_jk_B1ej

converges to 02E. Put

Bˆconv…B1[ [

j1ke_jk_B1ej† Now we prove that‰H…B†Š has property (LB¹).

In order to prove that‰H…B†Š has property (LB¹) by Vogt [9], it suffices to show that every continuous linear map T : ‰H…B†Šÿ!H…C† is bounded in a neighbourhood of 02 ‰H…B†Š. Consider the function f : Bÿ!H…C†gi- ven by

f…x†…† ˆT…x†…† forx2B; 2C

where_x2 ‰H…B†Šis the Dirac functional associated withx. We claim thatf is weakly holomophic, i.e.f 2H…B†for all2 ‰H…C†Š. Indeed, sinceE is a Schwartz - Frechet space so ‰H…B†Š is also a Schwartz - Frechet space.

Now let 2 ‰H…C†Š then T2 ‰‰H…B†ŠŠˆH…B† which gives a holo- morphic extension of f. For each s>0 consider h^sˆR^sf, where R^s : H…C† ÿ!H¹…2s† is the restriction map and ˆ f2C : jj<1g.

Thenh^s can be extended to a bounded holomorphic functionbh^s on a neigh- bourhoodV^s ofB inE. Takep1 such thatB‡UpV¹ and …eB1†holds for E, where U_p ˆ fx2E:kxk_p1g. Let V₁ˆB‡U_p and let g : …BC† [ …V₁†ÿ!Cbe given by

g…x; † ˆ f…x†…† for x2B; 2C bh¹…x†…† forx2V1; 2

Obviouslygis separately holomorphic in the sense of Siciak [7]. Byfwe denote the family of all finite dimensional subspacesPof E…B†, whereE…B†

denotes the Banach space induced byB. Put g_Pˆg

…B\PC†[…V1\P†

Since B\P and are not pluri - polar in V₁\P and C, respectively, by Nguyen and Zeriahi [5]g_Pis extended uniquely to a holomorphic functionbg_P on …V1\P† C. Since V1\E…B† ˆ [fV1\P:P2fg the family fbgP:P2fgdefines a Gaªteaux holomorphic functionbgon…V1\E…B†† C.

On the other hand,g is holomorphic onfx2B:kxk_B<1g , by Zorn's theorembg is holomorphic on…V₁\E…B†† C, whereV₁\E…B†is equipped with the topology ofE…B†.

Now we prove that bg is extended holomorphically to bg1 on WC, a neighbourhood of BC in EC such that bg1…Wr† is bounded for r>0. Letqp; d >0; C>0 be chosen such that (1) holds.

SinceBˆconv…B₁[ [

j1ke_jk_B1e_j†we have

ke_jk_B1ke_jk_B1; forj1 From the condition (1) we have

1 kejk_q

_1‡d

C

kejk_Bkejk^d_p …2†

Now let ˆ¹₂

C^1‡d1 e_ÿ1

. Given r>0; d>0 we can find s;D>0 such that

kk^1‡d_r Dkk_skk^d₁ …3†

for2H…C†, where

kk_kˆsupfj…z†j : jzj kg

Write the Taylor expansion of g : V1\E…B† ÿ!H…C†, the function asso- ciated tobg : …V₁\E…B†† Cÿ!Cat 02E…B†

g…x† ˆX¹

nˆ0

Png…x†

where

Png…x†…† ˆ 1 2i

Z

jtjˆ1

bg…tx; † t^n‡1 dt forx2V1\E…B†; 2C.

Sincebh^sis holomorphic at 02Efor everys>0 we infer thatP_ng… †…†is continuous on E for every . Let Pd_ng be the symmetricn-linear form asso- ciated withP_ng. We have

X

n0

jP_ng…x†…†j X

n0

X

j1;;jn1

je_j1…x†j ke_j1k_q je_jn…x†j ke_jnk_q kej1k_q kejnk_q jPdng…ej1; ;ejn†…†j

…4†

Using (2), (3) and (4) we get

X

n0

jPng…x†…†j X

n0

X

j1;;jn1

D^1‡d1 C^1‡dn je_j1…x†j kej1k_q je_jn…x†j kejnk_q kej1k^1‡d_B¹ kejnk^1‡d_B¹ kej1k^1‡d_p^d kejnk^1‡d_p^d kPd_ng…e_j1; ;e_jn†k^1‡d_s¹ kPd_ng…e_j1; ;e_jn†k^1‡d₁^d D^1‡d1 X

n0

C^1‡dn nⁿ

n!kPngk^1‡d_s;B¹kPngk^1‡d_1;p^d kxkⁿ_q D^1‡d1kgk^1‡d_Bs¹ kgk^1‡d_U^d_pX¹

nˆ0

C^1‡dn nⁿ

n! ⁿ<‡1 forx2U_p andjj<r.

Thusgis extended holomorphically to…U_qC† [ …V₁†. By the same argument, as above, g is extended holomorphically to g1 on V1C. Con- sider bg1 : V1ÿ!H_…C† associated with g1. By the same above argument it follows thatbg₁ is locally bounded. Hence there exists a neighbourhoodW of B in V₁ such that bg₁ is bounded. Define a continuous linear map S : ‰H¹…W†Šÿ!H…C†as

S…†…† ˆ…bg₁… ; ††

Since (1) holds forB1 it holds forB. This shows that Bis a set of unique- ness and we infer that span…B†is weakly dense in‰H…B†Š. Because‰H…B†Š

is reflexive span…B†is dense in‰H…B†Š, where :B! ‰H…B†Š is given by …x†…'† ˆ'…x†; x2B; '2H…B†. Now we have

T X^m

jˆ1

_jzj

…† ˆX^m

jˆ1

_jT…_zj†…† ˆX^m

jˆ1

_jf…z_j; †

ˆX^m

jˆ1

jbg1…zj; † ˆX^m

jˆ1

jS…zj†…† ˆS X^m

jˆ1

jzj

…†

for2C.

HenceS

‰H…B†Š ˆT and‰H…B†Š2 …LB¹†.

Proposition 3.4 is proved.

REFERENCES

1. S. Dineen,Complex Analysis in Locally Convex Spaces, Math. Stud. 57 (1981).

2. S. Dineen, R. Meise and D. Vogt,Characterization of nuclear Frechet spaces in which every bounded set is polar, B.S.M.F. 112 (1984), 41^68.

3. A. Grothendieck,Produits tensoriels topologiques et espaces nucle¨aires, Mem. Amer. Math.

Soc. 16 (1955).

4. Nguyen Van Khue and Bui Dac Tac,Extending holomorphic maps from compact sets in in- finite dimension, Studia. Math. 95 (1990), 263^272.

5. Nguyen Thanh Van and Zeriahi,Familles de polinoªmes presque partout borne¨es, Bull. Sc.

Math. (2) 107 (1983), 81^91.

6. P. Noverraz,Pseudoconvexite¨, convexite¨ polynomial et domaines d'holomorphie en dimension infinie, Math. Stud. 3 (1973).

7. J. Siciak,Separately analytic functions and envelopes of holomorphy of some lower-dimen- sional subsets ofCⁿ, Ann. Polon. Math. 22 (1969), 145^171.

8. J. Siciak,Weak analytic continuation from compact subsets ofCⁿ,in: Lecture Notes in Math.

364 (1974), 92^96.

9. D. Vogt,Fre¨chetraume, zwischen denen jede stetige lineare Abbildungbeschrankt ist,J. Reine Angew. Math. 345 (1983), 182^200.

10. D. Vogt,Subspaces and quotient spaces of (s),in: Functional Analysis: Surveys and Recent Results. K. D. Bierstedt and B. Fuschssteiner (eds.), North - Holland Math. Stud. 27 (1977), 167^187.

11. L. Waelbroeck,Weak analytic functions and the closed graph theorem, Proc. Conf. On In- finite Dimensional Holomorphy, Lecture Notes in Math. 364 (1974), 97^100.

12. V. P. Zahariuta,Isomorphism of spaces of analytic functions, Soviet Math. Dokl. 22 (1980), 631^634.

DEPARTMENT OF MATHEMATICS PEDAGOGICAL INSTITUTE HANOI TU LIEM - HANOI

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