(LB [infinity])-structure of spaces of germs of holomorphic functions

Publicacions Matem`atiques, Vol. 44 (2000), 177–192 (LB∞)-STRUCTURE OF SPACES OF GERMS OF HOLOMORPHIC FUNCTIONS Nguyen Dinh Lan Abstract We study the structure of spaces of germs of holomorphic functions on compact sets in Fr´echet spaces for (LB∞) as well as for (¯ Ω,˜ Ω). Introduction Let EbeaFr´echet space and let Kbe a compact subset in E.By H(K) we denote the space of germs of holomorphic functions on K equipped with the inductive limit topology. Some linear topological invariants, in particular those of the (Ω)-type for the strong dual [H(K)] of the space H(K), were investigated by several authors. For example, in the finite dimensional case, Zaharjuta proved that [H(K)]has (¯ Ω) if and only if Kis L-regular [17]. This problem, in the infinite dimensional case, has been considered already by R. Meise, D. Vogt and many others. Meise and Vogt have shown in [7] that [H(K)]has (Ω) for every compact subset Kin a nuclear Fr´echet space Eas long as Ehas (Ω). Recently, this result has been extended to the general case where Eis only Fr´echet by Nguyen Van Khue and Phan Thien Danh [10]. For the invariants (¯ Ω) and (˜ Ω), Meise and Vogt in [8] gave some necessary and sufficient conditions for the compact polydiscs ¯ Din a nuclear Fr´echet space having a Schauder basis such that [H(¯ D)]has (¯ Ω) and has (˜ Ω) respectively. The aim of the present paper is to study the invariant (LB∞) as well as (¯ Ω) and (˜ Ω) of [H(K)]in the case where Kis a balanced convex compact subset of a nuclear Fr´echet space E. It should be mentioned that this problem has been treated very recently by Le Mau Hai and Nguyen Van Khue [6] in the case where Eis a Fr´echet-Schwartz space having an absolute basis. Our main results are explained in Sections 2 and 3. Namely, in Section 2 by employing an important characterization of (LB∞) for Fr´echet spaces [15], we prove that if Bis a balanced convex compact subset of a Fr´echet space Ehaving (˜ ΩB) then [H(B)]has 178 N. Dinh Lan (LB∞) (Theorem 2.1). In Theorem 2.2, under the additional assumption that Ehas the bounded approximation property, we prove that Bis not pluripolar if [H(B)]has (LB∞). Combining this result and a characterization of (˜ ΩB) in terms of the non-pluripolarity of B[2] we also obtain a converse to Theorem 2.1 in the special case mentioned above. In Section 3, we prove in Theorem 3.1 that if Bis a balanced compact subset of a nuclear Fr´echet space having a Schauder basis then [H(B)] has either (¯ ΩB)or( ˜ ΩB) if and only if Ehas the same property. Finally, we note that the invariants of (DN)-type for spaces of entire functions of bounded type on (DF)-spaces were considered by several authors (for example [6], [10], ... ). 1. Preliminaries 1.1. Some linear topological invariants. Let EbeaFr´echet space with a fundamental system of semi-norms {•k}. For a subset Bof E, put u∗ B= sup {|u(x)|:x∈B}for u∈E. Write •∗ kfor B=Uk={x∈E:xk<1}. Using this notation we say Ehas the property (Ω) ⇔∀p∃q∀k∃C, d > 0•∗1+d q≤C•∗ k•∗d p. (Ω) ⇔∀p, d > 0∃q∀k>0∃C>0•∗1+d q≤C•∗ k•∗d p. (˜ Ω) ⇔∀p∃q,d > 0∀k∃C>0•∗1+d q≤C•∗ k•∗d p. (LB∞)⇔∀ρn↑∞∀p∃q ∀k∃nk,C >0 ∀u∈E∃nu∈[k;nk]u∗1+ρnu q≤Cu∗ nuu∗ρnu p. The above properties were introduced and investigated by Vogt (see [9]or[16] for (Ω) and [15] for the others). In [15] Vogt gave the following important characterization of (LB∞) for Fr´echet spaces. Vogt’s Theorem ([15, Satz 5.2]).For an arbitrary exponent sequence α=(αj)satisfying sup j≥1 αj+1 αj<∞, the following assertions are equivalent (i) Ehas (LB∞). (ii) Every continuous linear map from Einto Λ∞ ∞(α)is bounded on a zero-neighbourhood, where Λ∞ ∞(α)=(ξj)⊂C:(ξj)k:= sup |ξj|kαj<∞∀k≥1. (LB∞)-structure of spaces of germs 179 1.2. Holomorphic functions. Let E,Fbe locally convex spaces and Dan open subset in E. A function f:D−→ Fis called holomorphic if it is continuous and u◦fis Gˆateaux holomorphic for u∈F.ByH(D,F) we denote the space of F-valued holomorphic functions on D, equipped with the compact-open topology. When Fis omitted, it is understood to be the scalar field C, e.g. H(D)=H(D,C). Finally for each compact set Kin E,byH(K) we denote the space of holomorphic functions on K, equipped with the inductive topology, i.e. H(K) := lim ind U⊃KH∞(U) where Uranges over all neighbourhoods of Kand H∞(U) denotes the Banach space of bounded holomorphic functions on U. For the details concerning the holomorphic functions and the germs of holomorphic functions on compact sets in a locally convex space, we refer to the book of Dineen [1]. 2. The structure (LB ∞ ) Theorem 2.1. Let Ebe a nuclear Fr´echet space and Ba balanced convex compact subset in E. Assume that Ehas (˜ ΩB): (˜ ΩB):∀p∃q,d,C > 0•∗1+d q≤C•∗ B•∗d p. Then [H(B)]∈(LB∞). Note that in the definition of (˜ ΩB), by choosing qsufficiently large, we may assume that C=1. We need the following: Lemma 2.2. Let Eand Bbe as in Theorem 2.1. Then Bis a set of uniqueness. Here we say that the compact set Bis a set of uniqueness if for every f∈H(B), f|B= 0 implies f=0. Proof: First, since Ehas (˜ ΩB) by the hypothesis, it is easy to see that span Bis dense in E. Now given f∈H(B) with f|B= 0, consider the Taylor expansion of fat 0 ∈Bin a balanced convex neighbourhood W of Bin E: f(x)= n≥0 Pnf(x),x∈W, 180 N. Dinh Lan where Pnf(x)= 1 2πi  |λ|=δx>0 f(λx) λn+1 dλ for x∈E. Since Pnfare n-homogeneous polynomials and Pnf|B= 0, it follows that Pnf|span B= 0. By the continuity of Pnfand by span B=E,we have Pnf= 0 for n≥0. Thus f=0inWand hence Bis a set of uniqueness. Proof of Theorem 2.1: Since H(C)=Λ ∞ ∞(α) where α=(αj) with αj= jfor j≥1, by Vogt’s theorem it suffices to show that every continuous linear map T:[H(B)]−→ H(C) is compact. (i) Consider the function f:B−→ H(C) induced by T: f(x)(λ)=T(δx)(λ) for x∈B, λ ∈C, where δx∈[H(B)]denotes the Dirac functional associated to x∈B: ϕ, δx=ϕ(x) for ϕ∈H(B). It follows that fis weakly holomorphic, i.e. µ◦f∈H(B) for µ∈ [H(C)], because T(µ)∈[H(B)] ∼ =H(B). By Grothendieck’s factorization theorem [9], this yields that f:B−→ H∞(2∆), where ∆ is the open unit disc in C, is extended to a holomorphic function ˆ fon a neighbourhood Wof Bin E. Let g:(B×C)∪(Wׯ ∆) −→ Cgiven by g(x, λ)=f(x)(λ) for x∈B, λ ∈C ˆ f(x)(λ) for x∈W, λ ∈¯ ∆. Obviously gis separately holomorphic in the sense of Sciak [14], this means that g(x, ·) is holomorphic in λ∈Cfor every x∈Band g(·,λ)is too in x∈Wfor every λ∈¯ ∆. We denote by Fthe family of all finite dimensional subspaces P=0ofE(B), where E(B) is the Banach space spanned by B. For each P∈Fconsider gP=g|((B∩P)×C)∪((W∩P)ׯ ∆). Since B∩Pis the unit ball in Pand ¯ ∆ is not polar, by Nguyen Thanh Van-Zeriahi [11]gPis uniquely extended to a holomorphic function ˜gP on (W∩P)×C. The uniqueness implies that the family {˜gP:P∈F} defines a Gˆateaux holomorphic function ˜gon (W∩E(B)) ×C.On the other hand, since ˜gis holomorphic on (W∩E(B)) ×∆, Zorn’s theorem [1] implies that ˜gis holomorphic on (W∩E(B)) ×C. Consider the holomorphic function ˆg:(W∩E(B)) −→ H(C) associated to ˜g.We prove that ˆgcan be extended to a bounded holomorphic function on a neighbourhood of Bwith values in H(C). (LB∞)-structure of spaces of germs 181 (ii) The following is a modification of Meise-Vogt [8] and of Le Mau Hai [5]. Let •γ∞ γ=1 and {•k}∞ k=1 be two fundamental systems of seminorms of Eand H(C) respectively. Since H(C) has (DN)wehave ∃p∀q,d > 0∃k, C > 0•1+d q≤C•k•d p. Note that by replacing kwith some k>k, we always may assume that C= 1. Choose αsuch that Uα⊂Wand M(α, p) = sup ˆg(x)p:x∈Uα∩E(B)<∞. Let ωαfrom Einto Eα, the Banach space associated to •α, be the canonical map and A=ωα|E(B):E(B)−→ Eα. Since Eis nuclear, without loss of generality we may assume that E(B) and Eαare Hilbert spaces. Then, by [12, Proposition 8.6.6, p. 143], Acan be written in the form A(x)= j≥1 λjx, yjzj where λj>0∀j≥1, λ=(λj)∈s, the space of rapidly decreasing sequences, (yj) is a complete orthonormal system in E(B) and (zj)an orthonormal system in Eα. Since Ayj λj=zj∈ωα(Uα)∀j≥1, we have yj λj∈Uα∀j≥1. It follows that m  j=1 µj λjyj∈Uα,∀m≥1, where µj=δ jkand δ>0 is chosen such that    u∈Eα:u=∞  j=1 ξjzjand |ξj|<µ j∀j≥1  ⊂ωα(Uα) and δ∞  j≥1 1 jk≤1. 182 N. Dinh Lan We set χk∈E α:z∈Eα→ z,zkα,the scalar product in Eα. Then χk=1 ∀k≥1 and ∀k≥1A∗χk∗ B= sup x≤1|χkA(x)| = sup x≤1|A(x),z k| = sup x≤1|λkx, yk| =λk(by the Bessel inequality: |x, yk| ≤ x). (1) Now put ϕk=ω∗ αχk,(2) and choose βsuch that ∃d, C > 0•∗1+d β≤C•∗ B•∗d α.(3) For βsufficiently large, we can choose C=1. From (1)–(3) we have ϕk∗1+d β=ω∗ αχk∗1+d β≤A∗χk∗ Bχk∗d α≤λk∀k≥1. Hence ϕk∗ β≤(λk)1 1+d∀k≥1. Let h=ωpˆg. Since M(α, p)<∞and A(Uα∩E(B)) is dense in ωα(Uα), his holomorphically factorized through A:Uα∩E(B)−→ ˆ Uα by ˆ h:ˆ Uα−→ [H(C)]p, where ˆ Uαis the unit ball in Eα. This may be illustrated in the following diagram. Uα∩E(B)ˆg✲H(C) ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ h s ˆ Uα A ❄ ˆ h ✲[H(C)]p ωp ❄ . For each m=(m1,m 2,... ,m n,0,0,...)∈M, with M=m=(mj)∈NN:mj= 0 only for finitely many j∈N, (LB∞)-structure of spaces of germs 183 we put am=1 2πin |ρ1|=µ1 |ρ2|=µ2 ···  |ρn|=µn ˆ h(ρ1z1+ρ2z2+···+ρnzn) ρm+1 dρ where ρm+1 := ρm1+1 1ρm2+1 2...ρ mn+1 n, dρ := dρ1dρ2...dρ n, then amp≤M(α, p) µm∀m∈M. From the relation k  j=1 ρj λj yj∈Uα∩E(B)∀k≥1, we deduce that ˆ h  j≥1 ρjzj =ˆ hA   j≥1 ρj λj yj =ωpˆg  j≥1 ρj λj yj . On the other hand, by Cauchy’s theorem, we get am=1 2πin |ρ1|=λ1µ1 |ρ2|=λ2µ2 ···  |ρn|=λnµn ˆ h(ρ1z1+ρ2z2+···+ρnzn) ρm+1 dρ. It follows that am=1 2πin |ρ1|=λ1µ1 |ρ2|=λ2µ2 ···  |ρn|=λnµn ωpˆg( n  j=1 ρj λjyj) λm+1 ρ λm+1 dρ =ωp         1 λm1 2πin |θ1|=µ1 |θ2|=µ2 ···  |θn|=µn ˆg(θ1y1+θ2y2+···+θnyn) θm+1 dθ   bm         where θj=ρj λj∀j≥1. 184 N. Dinh Lan We have bmq≤N(q) λmµm∀m∈M, ∀q≥p, where N(q) = sup     ˆ h(x)  q:x=∞  j=1 ξjyjand |ξj|≤µj∀j≥1   <∞, because the set    x=∞  j=1 ξjyj:ξjyj∀j≥1   is compact in E(B). Since H(C) has (DN), for every q≥pand ¯ d=d δthere exists k≥q and C>0 such that •1+d q≤C•k•d p, where 0 <δ<1 is chosen such that ε:= t−1−t 1+ ¯ d>0 with t=1 2(1 + d). Again we may assume C= 1. Then S:=  m∈M rmbmq ∞  j=1 ϕj∗mj β≤ m∈M rmbmq ∞  j=1 (λj) mj 1+d = m∈M rmbmqλ2tm = m∈M rm λmbmq!t λtm bm1−t q ≤N(q)tN(k) 1−t 1+dM(α, p) (1−t)¯ d 1+d m∈M rmλm(t−1−t 1+d) µm(t+1−t 1+d+(1−t)d 1+d) ≤N(q)tN(k) 1−t 1+dM(α, p) (1−t)¯ d 1+d m∈M rmλm(t−1−t 1+d) µm. Since λ=(λj)∈s, the sequence "λε j µj#is in l1and hence for R=  j≥1"λε j µj#we have 2R>R>λε j µj for j≥1. (LB∞)-structure of spaces of germs 185 This implies 0<sup $λε j 2Rµj :j≥1%<1 2. We have S= m∈M rmbmq ∞  j=1 ϕj∗mj β ≤N(q)tN(k)1−t 1+ ¯ dM(α, p)(1−t)¯ d 1+ ¯ d m∈Mrλε µm =N(q)tN(k)1−t 1+ ¯ dM(α, p)(1−t)¯ d 1+ ¯ d ∞  j=1 1 1−rλε j µj <∞. Hence the form x→  m∈M bm j≥1 (ϕj(x))mj defines a bounded holomorphic function ˆ h1on δUβwith δ=1 4Rsuch that ˆ h1&&δUβ∩B=ˆg&&δUβ∩B, i.e. ˆ h1(z)(λ)=g(z,λ) for z∈δUβ∩B and λ∈¯ ∆. Since span B=E, by considering the Taylor expansion of ˆ h1(·)(λ)−g(·,λ)inz∈span Bat 0 ∈B, we get ˆ h1(z)(λ)=g(z,λ) for z∈δUβ∩Band λ∈¯ ∆. (iii) Consider the separately holomorphic function h1in the sense of Siciak [14]on(δUβ×C)∪(Wׯ ∆), induced by ˆ h1and g. By the same argument as in (i), h1is holomorphically extended to a function ¯ h1on W×C. Let ˆ ¯ h1:W−→ H(C) denote the holomorphic function associated to ¯ h1. Since Bis convex, balanced and the equality (ˆ h1−ˆg)&&δUβ∩B=0 holds, from the Taylor expansion of (ˆ h1−ˆg)|Bat 0 ∈Bit follows that ˆ ¯ h1|B=ˆg|B. (iv) Applying a similar argument as in (ii) to each point of W,it follows that ˆ ¯ h1is locally bounded. Thus, by shrinking W, without loss of generality, we may assume that ˆ ¯ h1(W) is bounded. Define the continuous linear map S:[H∞(W)]−→ H(C)as S(µ)(λ)=µ(ˆ ¯ h1(•)(λ)) for µ∈[H∞(W)]and λ∈C. 192 N. Dinh Lan [17] V. P. Zaharjuta, Isomorphism of spaces of analytic functions, Dokl. Akad. Nauk SSSR 255(1) (1980), 11–14. Department of Mathematics HoChiMinh City University of Education 280 An Duong Vuong, District 5 HoChiMinh City Vietnam E-mail address:[email protected] Primera versi´o rebuda el 8 de mar¸c de 1999, darrera versi´o rebuda el 24 de mar¸c de 2000.