Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces
The aim of this paper is to give the equivalence between the weak holomorphicity and the holomorphicity of Frechet-valued functions on compact polydiscs in (DFN)-spaces. Moreover, the relations between separately holomorphic functions and holomorphic functions on compact polydiscs in (DFN)-spaces...
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Інститут математики НАН України
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irk-123456789-1647862020-02-11T01:28:27Z Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces Pham Hien Bang Короткі повідомлення The aim of this paper is to give the equivalence between the weak holomorphicity and the holomorphicity of Frechet-valued functions on compact polydiscs in (DFN)-spaces. Moreover, the relations between separately holomorphic functions and holomorphic functions on compact polydiscs in (DFN)-spaces are also given. Мета цієї статті — встановити еквiвалентнiсть між слабкою голоморфністю та голоморфністю Фреше-значних функцій на компактних полідисках у (DFN)-просторах. Також наведено співвідношення між нарізно голоморфними функціями та голоморфними функціями на компактних полідисках у (DFN)-просторах. 2008 Article Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces / Pham Hien Bang // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1578–1584. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164786 517.9 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення |
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Короткі повідомлення Короткі повідомлення Pham Hien Bang Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces Український математичний журнал |
| description |
The aim of this paper is to give the equivalence between the weak holomorphicity and the holomorphicity
of Frechet-valued functions on compact polydiscs in (DFN)-spaces. Moreover, the relations between
separately holomorphic functions and holomorphic functions on compact polydiscs in (DFN)-spaces are
also given. |
| format |
Article |
| author |
Pham Hien Bang |
| author_facet |
Pham Hien Bang |
| author_sort |
Pham Hien Bang |
| title |
Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces |
| title_short |
Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces |
| title_full |
Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces |
| title_fullStr |
Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces |
| title_full_unstemmed |
Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces |
| title_sort |
fréchet-valued holomorphic functions on compact sets in (dfn)-spaces |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| topic_facet |
Короткі повідомлення |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/164786 |
| citation_txt |
Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces / Pham Hien Bang // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1578–1584. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT phamhienbang frechetvaluedholomorphicfunctionsoncompactsetsindfnspaces |
| first_indexed |
2025-07-14T17:22:11Z |
| last_indexed |
2025-07-14T17:22:11Z |
| _version_ |
1837643839232278528 |
| fulltext |
UDK 517.9
Pham Hien Bang (Thai Nguyen Univ. Education, Thai Nguyen, Vietnam)
FRECHET-VALUED HOLOMORPHIC FUNCTIONS
ON COMPACT SETS IN (DFN)-SPACES
ФРЕШЕ-ЗНАЧНI ГОЛОМОРФНI ФУНКЦIЇ
НА КОМПАКТНИХ МНОЖИНАХ У (DFN)-ПРОСТОРАХ
The aim of this paper is to give the equivalence between the weak holomorphicity and the holomorphicity
of Frechet-valued functions on compact polydiscs in (DFN)-spaces. Moreover, the relations between
separately holomorphic functions and holomorphic functions on compact polydiscs in (DFN)-spaces are
also given.
Мета цiєї статтi — встановити еквiвалентнiсть мiж слабкою голоморфнiстю та голоморфнiстю
Фреше-значних функцiй на компактних полiдисках у (DFN)-просторах. Також наведено спiв-
вiдношення мiж нарiзно голоморфними функцiями та голоморфними функцiями на компактних
полiдисках у (DFN)-просторах.
Introduction. Let E be a Frechet space (i.e., a complete metrizable locally convex
space) with a fundamental system of semi-norms {‖ · ‖k}. For each subset B of E, we
define ‖ · ‖∗B : E′ → [0,+∞] by
‖u‖∗B = sup
{
|u(x)| : x ∈ B
}
,
where u ∈ E′, E′ is the topological dual space of E.
Instead of ‖ · ‖∗Uk
we write ‖ · ‖∗k, where Uk =
{
x ∈ E : ‖x‖k 6 1
}
. Using this
notation, we say that E has the property
(DN) ∃p ∀q, d > 0 ∃k,C > 0
(DN) ∃p ∀q ∃k ∀d > 0 ∃C > 0
‖x‖1+d
q 6 C‖x‖k‖x‖d
p ∀x ∈ E.
(Ω) ∀p ∃q ∀k ∃d,C > 0
(Ω̃) ∀p ∃q, d > 0 ∀k ∃C > 0
‖u‖∗1+d
q 6 C‖u‖∗k‖u‖∗d
p ∀u ∈ E′.
Throughout this paper, if the Frechet space E has the property (DN) (respectively,
(DN), (Ω), (Ω̃)), then we write E ∈ (DN) (respectively, E ∈ (DN), E ∈ (Ω),
E ∈ (Ω̃)). The above properties have been introduced and investigated by Vogt [1] – [3].
In this paper, for all notions concerning the theory of holomorphic functions on
locally convex spaces and the theory of nuclear locally convex spaces, we refer readers
to the books of S. Dineen [4] and A. Pietsch [5]. However, for convenience of readers,
we recall some important notions which we use frequently here.
Let (Eα)α∈Γ be a collection of locally convex spaces. The locally convex space E
is the locally convex inductive limit of (Eα)α∈Γ and we write E = lim ind
α
Eα if for
each α in Γ, there exists a linear mapping iα: Eα → E such that E has the finest locally
convex topology for which each iα is continuous. A locally convex inductive limit of
normed spaces is called a bornological space.
Let X be a compact set in the Frechet space E. By H(X) we denote the space of
germs of holomorphic function on X. This space is equipped with the inductive topology
H(X) = lim ind
U↓X
H∞(U);
c© PHAM HIEN BANG, 2008
1578 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
FRECHET-VALUED HOLOMORPHIC FUNCTIONS ON COMPACT SETS IN (DFN)-SPACES 1579
here, for each neighborhood U ofX, we denote byH∞(U) the Banach space of bounded
holomorphic functions on U with the sup-norm
‖f‖U = sup
{
|f(z)| : z ∈ U
}
.
A locally convex space E is called to be quasi-Montel if every closed bounded subset
of E is compact. If p is a semi-norm on the vector space E, we set Ep = ̂(E/p−1(0), p)
(i.e., Ep is the Banach space obtained by factoring out the kernel of p and completing the
normed linear space (E/p−1(0), p). A locally convex space E is called to be Schwartz
space if for each continuous semi-norm p on E there exists a continuous semi-norm q on
E, q > p, such that the canonical mapping (i.e., the mapping induced by the indentity
on E) from Eq to Ep is compact. A linear mapping T between the Banach spaces E
and F is nuclear if there exists a sequence (λn)∞n=1 in l1, a bounded sequence (xn)∞n=1
in F, and a bounded sequence (ψn)∞n=1 in E′ such that Tx =
∑∞
n=1
λnψn(x)xn for
every x in E. A locally convex space E is called to be nuclear if for each continuous
semi-norm p on E there exists a continuous semi-norm q on E, q > p, such that the
canonical mapping from Eq to Ep is nuclear.
A sequence of vectors (en)∞n=1 in a locally convex space E is called a basis if for
each x in E there exists a unique sequence of scalars xn such that
x = lim
m→∞
m∑
n=1
xnen =
∞∑
n=1
xnen.
If the mapping Pm: E → E,
Pm
( ∞∑
n=1
xnen
)
=
m∑
n=1
xnen
are continuous for all m, the basis is called a Schauder basis. The Schauder basis
(en)∞n=1 is said to be absolute if for any semi-norm p on E there exists semi-norm q on
E such that
∞∑
n=1
|xn|p(en) 6 q
( ∞∑
n=1
xnen
)
for any
∑∞
n=1
xnen ∈ E.
A compact polydisc X in E′, the topological dual space of E, is called to be a
compact determining polydisc if every holomorphic function g on X such that g|X = 0
then g = 0 on a neighbourhood of X in E′.
Let E and F be locally convex spaces and let X be an open set in E. A function
f : X → F is called to be holomorphic on X if f is continuous and for every finite-
dimensional subspace G of E, f |G∩X is holomorphic function of several complex
variables. If the above-mentioned request holds for all u ◦ f, u ∈ F ′, the topological
dual space of F, then the function f is said to be weakly holomorphic on X.
ByH(X,F ) (respectively,Hw(X,F )) we denote the vector space of all holomorphic
(respectively, weakly holomorphic) functions on X with values in F. The aim of the
present paper is to find some conditions for which
(A) H(X,F ) = Hw(X,F ).
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
1580 PHAM HIEN BANG
This problem has been interested by some authors [6] – [10]. In [7], L. M. Hai has
shown that F ∈ (DN) if and only if (A) holds for every L̃-regular compact set X in a
Frechet space E, where a compact subset X in a Frechet space E is called L̃-regular if
[H(X)]′ ∈ (Ω̃). Moreover, he also has shown that for a compact polydisc X in a dual
space of a nuclear Frechet space E with a basis, (A) holds for every Banach space F if
and only if E ∈ (DN).
In this paper, we shall prove the following theorems:
Theorem A. Let E be a Frechet nuclear space with a basis having a continuous
norm. Then the following statements are equivalent:
(i) E has the property (DN);
(ii) H(X,F ) = Hw(X,F ) and H(X) is quasi-Montel for every compact determi-
ning polydisc X in E′ and for every Frechet space F ∈ (DN).
To state the second theorem, we give the following notion. Let X be a compact set
in a locally convex space E and let f: X → H(F ) be a continuous function with values
in H(F ). The function f is called to be separately holomorphic if δx ◦ f ∈ H(X) for
every x ∈ F, where δx : H(F ) → C is given by
δx(ϕ) = ϕ(x) for each ϕ ∈ H(F ).
By Hδ(X,H(F )) we denote the vector space of separately holomorphic functions on X
with values in H(F ).
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′. By
Hb(E′) we denote the metric locally convex space of entire functions of bounded type
on E′ equipped with the topology of the convergence on bounded sets in E′.
Theorem B. Let E be a Frechet nuclear space with a basis and have a continuous
norm. Then the following statements are equivalent:
(i) E has the property (DN);
(ii) H(X,Hb(F ′)) = Hδ(X,Hb(F ′)) holds for every compact determining polydisc
X in E′ and either every Frechet – Schwartz space F ∈ (DN) having an absolute basis
or every Banach space F.
Theorem C. Let E be a nuclear Frechet space with a basis and have a continuous
norm. Then
H(X,F ′) = Hw(X,F ′)
holds for every compact determining polydisc X in E′ and for every Frechet space F if
and only if E ∈ (DN).
1. Proof of Theorem A. For the proof of Theorem A, we need the following two
lemmas:
Lemma 1 [6]. Let B be a Banach space and let H(OB) denote the space of germs
of holomorphic functions at O in B. Then
[
H(OB)
]′
β
∈ (Ω).
Lemma 2 [11]. Every continuous linear map from a Frechet space E ∈ (Ω) into
a Frechet space F ∈ (DN) can be factorized through a Banach space. This means
that there exists a continuous semi-norm ρ on E and a continuous linear map g :
Eρ → F, where Eρ is the Banach space associated to the continuous semi-norm such
that f = g ◦ Φρ, Φρ : E → Eρ is the canonical quotient map.
We now prove Theorem A.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
FRECHET-VALUED HOLOMORPHIC FUNCTIONS ON COMPACT SETS IN (DFN)-SPACES 1581
Sufficiency. Assume that E ∈ (DN), F ∈ (DN), and X is a determining compact
polydisc in E′ of the form
X =
{
ω = (ωn) ∈ E′ : sup
n>1
|ωnαn| 6 1
}
,
where {αn}n>1 is a sequence of positive numbers. Since H(X) is regular [4], H(X) is
quasi-Montel. It suffices to show thatHw(X,F ) ⊂ H(X,F ). Let f ∈ Hw(X,F ). Since
X is compact, αn 6= 0 for every n > 1. Note that H(X) is regular because E ∈ (DN)
[4] and since X is determinating then we can consider the linear map f̂ : F ′ → H(X)
given f̂(u) = ûf , a holomorphic extension of uf to a neighbourhood of X.
By [4], on a neighbourhood (depending on u) of X, we have
f̂(u)(ω) =
∑
m∈N(N)
bm(u)ωm,
where
bm(u) =
1
(2πi)n
∫
|λ1|= 1
|α1|
. . .
∫
|λn|= 1
|αn|
f̂(u)(λ1e
∗
1 + . . .+ λne
∗
n)
λm1+1
1 . . . λmn+1
n
dλ
and
N (N) =
{
m = (mn)∞n=1; mn is a nonnegative integer for all n
and mn = 0 for all n sufficiently large
}
,
{ej}j>1 and {e∗j}j>1 are bases of E and E′, respectively.
We check that bm(u) are continuous on F ′. Fix m ∈ N (N) and put
Xm = X ∩ span{e∗1, . . . , e∗n} =
{
(ω1, . . . , ωn) : |ωi| 6
1
|αi|
, i = 1, n
}
.
Consider fm = f |Xm . By the hypothesis, fm ∈ Hw(Xm, F ). By [7] and by the
L̃-regularity of Xm in span {e∗1, . . . , e∗n}, we have fm ∈ H(Xm, F ). Thus, there
exists a neighbourhood Vm of Xm in span {e∗1, . . . , e∗n} for which fm is extended
to a holomorphic function f̂m: Vm → F. Hence,
bm(u) =
1
(2πi)n
∫
|λ1|= 1
|α1|
. . .
∫
|λn|= 1
|αn|
f̂(u)(λ1e
∗
1 + . . .+ λne
∗
n)
λm+1
dλ =
=
1
(2πi)n
∫
|λ1|= 1
|α1|
. . .
∫
|λn|= 1
|αn|
f̂m(u)(λ1e
∗
1 + . . .+ λne
∗
n)
λm+1
dλ, (1)
where λm+1 = λm1+1
1 . . . λmn+1
n is continuous on F ′.
We now prove that f̂: F ′
bor → H(X) is continuous, where F ′
bor means that the space
F ′ is equipped with the bornological topology. Take arbitrary µ ∈ [H(X)]′ ∼= H(U)
[12], where
U =
{
z = (zn) ∈ E : sup
∣∣∣∣ zn
αn
∣∣∣∣ < 1
}
is an open polydisc in E.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
1582 PHAM HIEN BANG
By [4], we can write
µ(z) =
∑
N(N)
am(µ)zm, z ∈ U,
and 〈
f̂(u), u
〉
=
∑
N(N)
bm(u)am(µ). (2)
From
sup
{∣∣∣∣∣∑
J
bm(u)am(µ)
∣∣∣∣∣ : J ⊂ N (N), J is finite
}
<∞,
for u ∈ F ′, we infer that
sup
{∣∣∣∣∣∑
J
bm(u)am(µ)
∣∣∣∣∣ : J ⊂ N (N), J is finite, u ∈ B′
}
<∞
for all bounded sets B′ ⊂ F ′.
This implies the continuity of f̂ : F ′
bor → H(X). By the regularity of H(X), there
exists an increasing sequence of bounded sets {Bk} in E such that f̂ maps continuously
F ′
bor into lim ind
k
H∞(B0
k). Choose {εk} ↓ 0 such that B =
∞⋃
k=1
εkBk is bounded in
E. Let P and Q denote the space E′/ker‖ ‖∗B equipped with the topology generated by
{‖ ‖∗Bk
} and ‖ ‖∗B respectively. Note that
id :
(
E′/ker‖ ‖∗B , Q
)
→
(
E′/ker‖ ‖∗B , P
)
is continuous. It follows that
lim ind
k
H∞(B0
k) = H(OP )
which is continuously embedded into H(OQ). Lemmas 1 and 2 imply that there exists a
neighbourhoodW of O in F ′
bor and a neighbourhood U ofX such that f̂(W ) ⊂ H∞(U).
Define a holomorphic function
g : U → F
given by
g(x)(u) = f̂(u)(x)
for x ∈ U, x ∈ F ′. We have
g(x)(u) = f̂(u)(x) = f(u)(x)
for x ∈ X, u ∈ F ′. Thus, g|X = f and f is extended holomorphically to U.
Necessity. Let {ei} be a basis of E with the dual basis {e∗j} ⊂ E. Since E has a
continuous norm, there exists an open polydisc in E of the form
U =
z =
∞∑
j=1
zjej ∈ E : sup
j>1
|zj |pj < 1
,
where pj > 0 for all j > 1.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
FRECHET-VALUED HOLOMORPHIC FUNCTIONS ON COMPACT SETS IN (DFN)-SPACES 1583
Hence,
X = UM =
ω =
∞∑
j=1
ωje
∗
j : |ωj | 6 pj for j > 1
= p1∆× Y
is a compact determining polydisc in E′; here ∆ =
{
ω1 ∈ C : |ω1| 6 1
}
and Y =
=
{
(ωj)j>2: |ωj | 6 pj for j > 2
}
.
Indeed, given f ∈ H(X) such that f |X = 0. Take a convex neighbourhood W of
X in E′ such that f ∈ H(W ). For each m > 1, put L = span {e∗1, e∗2, . . . , e∗m} and
consider
V =
∑
16j6m
λjpje
∗
j :
m∑
j=1
|λj | 6 1
.
Note that V is a neighbourhood of 0 ∈ L which is contained in X. By the hypothesis,
f |V = 0. Thus, f |W∩L = 0. Hence,
f |W∩span{e∗j : j>1} = 0.
The density of W ∩ span {e∗j : j > 1} in W and the continuity of f imply that f |W = 0.
First, we show that H(X) is regular. Since H(X) is quasi-Montel, H(X) is quasi-
reflexive.
Now given a balanced convex bounded set A in H(X). Consider the normed space
E1 = H(X)(A) spanned by A and the function f: X → E′
1 given by f(x)(σ) = σ(x)
for x ∈ X, σ ∈ E1.
Since H(X) = H(X)′′, we infer that f is weakly holomorphic. By the hypothesis,
f can be extended to a bounded holomorphic function f̂ on a neighbourhood V1 of X
in E′. From the relation
σ(x) = f(x)(σ) = f̂(x)(σ)
for every x ∈ X and σ ∈ A and from the uniqueness of X it follows that A is
contained and bounded in H∞(V1). Thus, H(X) is regular and hence, by [4], H(U) is
bornological. Since Lb(F, H(p1∆)) ∼= F ′⊗̂ΠH(p1∆) contained in
H(W1)⊗̂ΠH(p1∆) ∼= H(W1)⊗̂εH(p1∆) ∼= H(p1∆×W1) = H(U)
is a complemented subspace of one, where E = Ce1 ⊗ F, U = p1∆ ×W1, it follows
that Lb
(
F, H(p1∆)
)
is bornological. By [3] (Theorem 4.9), F and hence, E ∈ (DN).
2. Proof of Theorem B. We need the following lemma:
Lemma 3 [13]. Let E, F be Frechet spaces with E ∈ (Ω), F ∈ (DN). Assume
that F is a Schwartz space with an absolute basis. Then every E′-valued holomorphic
function on F ′ is factorized through a Banach space.
We now give the proof of Theorem B.
Sufficiency. Given f ∈ Hδ(X,Hb(F ′)).
(i) First assume that F is a Frechet – Schwartz space with an absolute basis and
F ∈ (DN). Consider the map f̂ : F ′ → H(X) given by f̂(u) = δ(u) ◦ f for u ∈ F ′.
We use again the notations of Theorem A. By (1) and (2), for each u ∈ [H(X)]′, the set∑
j
bm(·)am(µ) : J ⊂M,J is finite
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
1584 PHAM HIEN BANG
is bounded in Hb(F ′) and hence, it is relatively compact in Hb(F ′). This yields that
the function f̂ : F ′ → H(X) is weakly holomorphic. Since F ′ is a DFS-space, it
implies that f̂ is holomorphic. Now, as in the proof of Theorem A, by Lemma 3, f̂ is
holomorphically factorized through a Banach space. By the regularity of H(X), as in
Theorem A, we can extend f to an element of H
(
X,Hb(F ′)
)
.
(ii) Let F be a Banach space. As in (i), f̂: F ′ → H(X) is a holomorphic function of
bounded type. Again, by the regularity ofH(X), it implies that f can be holomorphically
extended to an element of H(X,Hb(F ′)).
Necessity. As in the proof of Theorem A, it suffices to show that H(X) is regular.
Given A a balanced convex closed bounded set in H(X). As in Theorem A, consi-
der the function f : X → H(X)(A). Since H(X)(A) ⊂ Hb([H(X)(A)]′), f ∈
∈ Hδ
(
X,Hb
(
[H(X)(A)]′
))
. By the hypothesis, f is extended to a bounded holomorphic
function f̂ : V → Hδ([H(X)(A)]′). Since X is an uniqueness set, we may assume that
f̂ : V → H(X)(A) is holomorphic. This yields that A is contained and bounded in
H∞(V ). The theorem is proved.
3. Proof of Theorem C. Given f ∈ Hw(X,F ′). Since f(X) is bounded, we can
find a neighbourhood V of 0 ∈ F such that f(X) is contained and bounded in F ′(V 0).
Since F ′(V 0) ∼= (F̃ (V ))′, where F̃ (V ) is the Banach space associated to V, it implies
that f: X → F ′(V 0) ⊂ Hb(F̃ (V )) and f ∈ Hδ(X,Hb(F̃ (V ))). Theorem B yields that
f ∈ H(X,Hb(F̃ (V ))) and as in the proof of Theorem B we infer that f ∈ H(X,F ′).
The necessity follows from the proof of Theorem A.
Acknowledgment. The author would like to thank Prof. Le Mau Hai for suggesting
the problem and for useful comments during the preparation of this work.
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Received 02.04.07,
after revision — 24.03.08
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 11
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