Structure of spaces of germs of holomorphic functions

Publicacions Matem`atiques, Vol 41 (1997), 467–480. STRUCTURE OF SPACES OF GERMS OF HOLOMORPHIC FUNCTIONS N. Van Khue and P. Thien Danh Abstract Let Ebe a Frechet (resp. Frechet-Hilbert) space. It is shown that E∈(Ω) (resp. E∈(DN)) if and only if [H(OE)]∈(Ω) (resp. [H(OE)]∈(DN)). Moreover it is also shown that E∈(DN) if and only if Hb(E)∈(DN). In the nuclear case these results were proved by Meise and Vogt [2]. 1. Preliminaries 1.1. Let Kbe a compact set in a Frechet space E.ByH(K)we denote the space of germs of holomorphic functions on K. This space is equipped with the inductive topology H(K) = lim ind U↓KH∞(U). Here for each neighborhood Uof K,byH∞(U) we denote the Banach space of bounded holomorphic functions on Uwith the sup-norm fU= sup{|f(z):z∈U}. 1.2. Let Edenote the strong dual space of a Frechet space E.A holomorphic function on Eis said to be of bounded type if it is bounded on every bounded set in E.ByHb(E) we denote the metric locally convex space of entire functions of bounded type on Eequipped with the topology the convergence on bounded sets in E. For more details concerning holomorphic functions on locally convex spaces we refer to the book of Dineen [1]. 468 N. Van Khue, P. Thien Danh 1.3. Assume the topology of Eis defined by an increasing fundamental system of seminorms {· k}∞ k=1. For each subset Bof E, define the generalized seminorm · ∗ B:E→[0,+∞], by u∗ B= sup{|u(x)|:x∈B}. Write · ∗ qfor B=Uq={x∈E:xq≤1}. Using this notation define Eto have the property (DN): ∃p∀q∃k, C > 0:2 q≤Ckp. (Ω) : ∀p∃q∀k∃d, C > 0:· ∗1+d q≤C· ∗ k· ∗d p. The properties (DN), (Ω) and the other many properties were introduced and investigated by Vogt (see, for example, [7], [8], etc.). In [8]he has proved that E∈(DN) (resp. E∈(Ω)) if and only if Eis isomorphic to a subspace (a quotient space) of the space Bˆ ⊗πsfor some Banach space B, where sis the space of rapidly decreasing sequences. The following three theorems are proved in the present paper. Theorem 1. Let Ebe a Frechet space. Then the following are equivalent (i) E∈(Ω) (ii) [H(K)]∈(Ω) for some non-empty compact set Kin E. (iii) [H(K)]∈(Ω) for all compact sets Kin E. Theorem 2. Let Ebe a Frechet-Hilbert space. Then E∈(DN)if and only if [H(OE)]∈(DN). Theorem 3. Let Ebe a Frechet space. Then E∈(DN)if and only if Hb(E)∈(DN). The proofs of Theorems 1, 2 and 3 are presented in Sections 2, 3 and 4 respectively. Structure of spaces of germs of holomorphic functions 469 2. Proof of Theorem 1 2.1. Lemma. Let Ebe a Frechet space. Then E∈(Ω) if and only if Eis isomorphic to a subspace of Bˆ ⊗πsfor some Banach space B. Proof: Suppose Eis isomorphic to a subspace of the space Bˆ ⊗πs where Bis some Banach space. Then E is isomorphic to a quotient space of (Bˆ ⊗πs)∼ =Bˆ ⊗πsand hence E ∈(Ω). This implies that E∈(Ω) since u∗ k= sup{|v(u)|:v∈U00 k⊂E}for u∈E. Conversely assume that E∈(Ω). Consider the canonical resolution 0−→ E−→  k≥1 Ek R −→  k≥1 Ek−→ 0 as constructed by Palamodov [4], where for each k≥1, Ekstands for the Banach space associated to k. Since Eis isomorphic to a quotient space of Bˆ ⊗πswith Bis some Banach space [8], Eis quasinormable. Hence we may assume that every bounded set in Ek+1 can be approximated by a bounded set in Ek+2 under the canonical map Ek+2 →Ek. It follows from [4] that every bounded set in k≥1Ekis the image of a bounded set in k≥1Ekunder R. By modifying the argument in [8] we imply that Eis isomorphic to a quotient space of Bˆ ⊗πsfor which Eis isomorphic to a suspace of (Bˆ ⊗πs)∼ =Bˆ ⊗πs. The following lemma is an immediate consequence of the preceding lemma. 2.2. Lemma. Let Ebe a Frechet space. Then E∈(Ω) if and only if E ∈(Ω). 2.3. Lemma. Let Bbe a Banach space. Then [H(OBˆ ⊗πs)]∈(Ω). Proof: Let {ej}be the canonical basis of swith the dual basis {e∗ j}of s. Since sis nuclear without loss of generality we may assume that 470 N. Van Khue, P. Thien Danh δp= j≥1 e∗ j∗ p+1ejp<1/e for p≥1. For each p≥1, put |f|p+1 = sup   1 p+1n j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)| ×e∗ j1∗ p+1 ...e∗ jn∗ p+1 :u1,... ,u n∈W, n ≥0   for f∈H ∞(conv(W⊗Up)), where Wis the unit ball of Band f(ω)= n≥0 Pnf(ω),ω∈conv(W⊗Up) is the Taylor expansion of fat 0 ∈Bˆ ⊗πs. Since |f|p+1 = sup  1 p+1n j1,... ,jn≥1 Pnfu1⊗ej1 ej1p ,... ,u n⊗ejn ejnp ×e∗ j1∗ p+1 ...ejnp...e∗ j1∗ p+1ej1p:u1,... ,u n∈W, n≥0   ≤Cpfconv(W⊗Up) for f∈H ∞(conv(W⊗Up)), where Cp= sup  δp p+1nnn n!:n≥0<∞, it follows that | |p+1 is continuous on H∞(conv(W⊗Up)). Structure of spaces of germs of holomorphic functions 471 On the other hand, we have fconvW⊗Up+1 p+2 = sup   f 1 p+2 k≥1 λkuk⊗vk  :uk∈W, vk∈Up+1, k≥1 |λk|≤1   ≤sup    n≥0p+1 p+2n k1,... ,kn≥1 |λk1|...|λkn| j1,... ,jn≥11 p+1n ×| Pnf(uk1⊗ej1,... ,u kn⊗ejn)||e∗ j1vk1|...|e∗ jn(vkn)| :uk∈W, vk∈Up+1, k≥1 |λk|≤1   ≤sup    n≥0p+1 p+2n k1,... ,kn≥1 |λk1|...|λkn| j1,... ,jn≥11 p+1n ×| Pnf(uk1⊗ej1,... ,u kn⊗ejn)|e∗ j1∗ p+1 ...e∗ jn∗ p+1 :uk∈W,  k≥1 |λk|≤1   ≤sup    n≥0p+1 p+2n k1,... ,kn≥1 |λk1|...|λkn|f|p+1 : k≥1 |λk|≤1   ≤  n≥0p+1 p+2n |f|p+1. Hence H(OBˆ ⊗πs)∼ =lim ind[H∞(conv(W⊗Up)) : | |p+1]. Given p≥1, choose q≥psuch that ∀k∃C, d > 0:e∗ j∗1+d q≤Ce∗ j∗ ke∗ j∗d p∀j≥1. 472 N. Van Khue, P. Thien Danh Since ∗ q≤∗ p, the above inequality holds for every d≥d. Hence we may assume that Ckpd≤q1+d, then |f|1+d q= sup   1 qn j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)| ×e∗ j1∗ q...e∗ jn∗ q:u1,... ,u n∈W, n ≥0   1+d ≤sup   1 Ckpdn 1+d j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)| ×Cn 1+de∗ j1∗1/1+d k...e∗ jn∗1/1+d k ×e∗ j1∗d/1+d p...e∗ jn∗d/1+d p:u1,... ,u n∈W, n ≥0   1+d ≤sup   (1/k)n j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)| ×e∗ j1∗ k...e∗ jn∗ k:u1,... ,u n∈W, n ≥0   ×sup   (1/p)n j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)| ×e∗ j1∗ p...e∗ jn∗ p:u1,... ,u n∈W, n ≥0   d =|f|k|f|d p for f∈H(OBˆ ⊗πs). Since Bˆ ⊗πsis quasinormale, according to Mujica [3] there exists a Frechet space Fsuch that F∼ =H(OBˆ ⊗πs). Combining this fact together with the inequality | |1+d≤||k| |d pon H(OBˆ ⊗πs), Structure of spaces of germs of holomorphic functions 473 by virtue of Lemma 2.2 we obtain [H(OBˆ ⊗πs)]∼ =F has (Ω). Now we are able to prove Theorem 1. (iii) →(ii) is trivial. (ii) →(i). Fix x0∈K. Then the form ϕ→ϕ(x0) defines a left inverse of the canonical map E→H(K). Hence E ∈(Ω), this implies that E∈(Ω). (i) →(iii). Assume that E∈(Ω). By Vogt [5] there exists a continuous linear map Rfrom Bˆ ⊗πsonto Efor some Banach space B. Let {Wk}be a neighbourhood basis of 0 ∈Bˆ ⊗πs. Then {Vk=R(Wk)} forms a neighbourhood basis of 0 ∈E. Lemma 2.3 gives ∀p∃q∃C, d > 0∀f∈H ∞(Wp):f1+d Wq≤CfWkfd Wp. Thus (∗)g1+d Vq≤gVkgd Vp∀g∈H ∞(Vp). Next let Kbe an arbitrary compact set in E. From (∗) we deduce that g1+d K+Vq≤CgK+Vpgd K+Vp∀f∈H ∞(K+Vp). According to Mujica [3] there exists a Frechet space Fverifying F∼ = H(K) since Eis quasinormale, invoke Lemma 2.2 we conclude that [H(K)]∼ =F ∈(Ω). 3. Proof of Theorem 2 3.1. Lemma. Let Ebe a Frechet-Hilbert space with E∈(DN). Then Eis isomorphic to a subspace of the space l2(I)ˆ ⊗πsfor some index set. Proof: Choose an index set Isuch that Eis isomorphic to a subspace of [l2(I)]N. Consider the exact sequence of nuclear Frechet spaces 0→s→s→ω→0 constructed by Vogt [8]. By tensoring this sequence with l2(I) we get the exact sequence of Frechet-Hilbert spaces [8], 0→l2(I)ˆ ⊗πs→l2(I)ˆ ⊗πsq →[l2(I)]N→0 474 N. Van Khue, P. Thien Danh Let ˜ E=q−1(E). Since 0→l2(I)ˆ ⊗πs→ˆ Eq →E→0 is a exact sequence of Frechet-Hilbert spaces in which l2(I)ˆ ⊗πs∈(Ω) and E∈(DN)byVogt[9]qhas a right inverse. Hence Eis isomorphic to subspace of ˜ Eand hence of l2(I)ˆ ⊗πs. 3.2. Lemma. Let Bbe a Banach space. Then [H(OBˆ ⊗πs)]∈(DN). Proof: Let Wdenote the unit ball of B. Write the Taylor expansion of each f∈H(OBˆ ⊗πs)at0∈Bˆ ⊗πs f(ω)= n≥0 Pnf(ω). Formally we have f  k≥1 λkuk⊗vk = n≥0 k1,... ,kn≥1 λk1...λ kn × j1,... ,jn≥1 Pnf(uk1⊗ej1,... ,u kn⊗ejn)e∗ j1(vk1)...e ∗ jn(vkn) for ω= k≥1 λkuk⊗vk∈Bˆ ⊗πs. For each p≥1 as in Theorem 1 put |f|p= sup    1 pn j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)|(j1...j n)−p :n≥0,u 1,... ,u n∈W   and Fp={f∈H(OBˆ ⊗πs):|f|p<+∞}. Then H(OBˆ ⊗πs)∼ =lim ind pFp. Structure of spaces of germs of holomorphic functions 475 In order to prove that [H(OBˆ ⊗πs)]∈(DN) we check that (2) ∀q∃k, C > 0:Wq⊂CsW1+1 sWk∀s>0 where for each q≥1 put Wq={f∈Fq:|f|q<1}. Obviously (4) holds for 0 <s≤1. Let s>1. Choose k=q3. We have (3) sup    1 kn j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)|(j1...j n)−k :n≥log s log k q ,u 1,... ,u n∈W   + sup      1 kn (j1...jn≥s 1 k−q | Pnf(u1⊗ej1,... ,u n⊗ejn)|(j1...j n)−k :n≥0,u 1,... ,u n∈W     ≤sup    1 qn j1,... ,jn≥1 | Pnf(u1⊗ej1,... ,u n⊗ejn)|(j1...j n)−q :n≥0,u 1,... ,u n∈W   ≤sup q kn:n≥log s log k q+ sup{(j1...j n)q−k:(j1...j n)≥s1 k−q} ≤2 sfor f∈Wq