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52 0016–2663/04/3801–0052 c
2004 Plenum Publishing Corporation
Functional Analysis and Its Applications, Vol. 38, No. 1, pp. 52–64, 2004
Translated from Funktsional
nyi Analiz i Ego Prilozheniya, Vol. 38, No. 1, pp. 65–80, 2004
Original Russian Text Copyright c
by B. N. Khabibullin
Closed Submodules of Holomorphic Functions
with Two Generators∗
B. N. Khabibullin
Received July 4, 2002
Abstract. Let Ibe a closed submodule over a polynomial ring in a space of holomorphic functions
on a domain in the complex plane. We establish sufficient conditions under which Iis generated
by two functions or two special submodules. As a corollary, it follows from these results that if
an invariant subspace W⊂C∞(a, b) (with respect to the differentiation operator) admits spectral
synthesis, then it is the solution space of a system of two homogeneous convolution equations.
Key words: spaces of holomorphic functions, module over the ring of polynomials, local description
of closed submodules, finitely generated submodule, spectral synthesis, convolution equation.
Introduction
In this paper, we mainly use the terminology in [1–3] and, for the notions of functional analysis,
in [4].
In what follows, Ω is a domain in the complex plane C,H(Ω) is the space of all holomorphic
functions in Ω, and P⊂H(Ω) is some separable locally convex space over C, which, for brevity,
will be referred to as a space. Everywhere below, it will be assumed that the topology of the space
Pmajorizes the topology of pointwise or simple convergence on Ω.
If the space Pis a topological module over the polynomial ring C[z], z∈C,oratopological
algebra, then, accordingly, Pwill be simply called a module or an algebra for brevity. Thus, in what
follows, the space Pis a module if it is closed and continuous with respect to the multiplication
by the independent variable z.
A subspace Iin the module Pis a submodule ∗∗ over C[z]ifpf ∈Ifor all p∈C[z]andf∈P.
In particular, if Pis an algebra containing all polynomials, then every ideal in Pis a submodule.
A closed submodule Iin the module Pis said to be (topologically) n-generated,n∈N,if
there is a set of nelements g1,...,g
n∈Isuch that Icoincides with the closure in Pof the set
of elements of the form p1g1+···+pngn, where p1,...,p
n∈C[z]. In this case, g1,...,g
nare the
functions generating the submodule I, and the submodule Iitself is denoted by ¯
I(g1,...,g
n). A
submodule is said to be principal if it is 1-generated. More generally, a closed submodule I⊂Pis
said to be (topologically) generated by submodules I1,...,I
nif it is the closure of the set of elements
of the form i1+···+in, where i1∈I1,...,i
n∈In.
In this paper, on the basis of the results of I. F. Krasichkov-Ternovskii’s foundational investi-
gations on the problem of local description of closed submodules in spaces of holomorphic function
of one variable, we establish conditions under which a closed submodule Iin the module Pis
2-generated or is generated by two special submodules.
In the case Ω = D, where Dis a unit disk, the topics in the present paper are close to the
numerous descriptions (that appeared after the publication of A. Beurling’s famous theorem on the
description of z-invariant subspaces in the Hardy class H2)ofalgebrasinH(D) whose every closed
ideal is principal. (See the surveys by Nikolskii [5, Secs. 7 and 9–13] and [6] and the references there.)
Among the series of works in this investigation direction by F. A. Shamoyan and his disciples, we
only mention [7, 8].
∗This work was supported by Russian Foundation for Basic Research Grant No. 00-01-00770 and partially by
Grant No. 02-01-00030.
∗∗ In his connection, the term z-invariant or translation-invariant subspace is also widespread [5, 6].
53
In weighted algebras of entire functions determined by radial majorants, every closed ideal is
almost always 2-generated (see [9]) and, for slowly increasing majorants, even 2-generated in the
algebraic sense [10]. In this situation, there can be closed ideals that are not principal, e.g., in
algebras of integer-order entire functions of finite type. Abuzyarova (see [11–13]) established re-
cently that, in the case of weighted modules of entire functions singled out by some conditions
on the indicator, every closed submodule admitting a local description (for the related definitions,
see Sec. 1) is 2-generated. Our results include the above-mentioned situations and also some very
wide classes (see Sec. 5) of modules of holomorphic functions in domains belonging to C. Applica-
tions of the main results in this paper (Theorems 1 and 2) to the problem of spectral synthesis are
discussed in Sec. 5 for the simple, but rather meaningful case of the space C∞(a, b) of infinitely
differentiable functions on an interval (a, b)⊂R. By Theorem 4, every differentiation-invariant
closed subspace W⊂C∞(a, b) such that the linear span of all exponential monomials contained
in Wis dense in it can be defined as a solution space of at most two homogeneous convolution
equations.
The unquestionable connection of the problem under consideration with corona theorems and
with weakly invertible elements is not discussed here. The simpler case in which Pis an algebra
and in which only closed ideals instead of closed submodules are considered is touched upon in
this paper merely episodically since it was considered rather completely in [9]. Some of the results
obtained in [9] and in this paper were partially announced earlier. (See [14, Theorems A and M].)
The author is grateful to I. F. Krasichkov-Ternovskii for some valuable recommendations that
facilitated the adequacy between the terminology and the newly introduced notions and also to
S. V. Popenov for useful information.
1. Main Notions and Results
In what follows, a divisor on Ω is understood either as a nonnegative integer-valued function Λ
with a support supp Λ = {λn}⊂Ω which is a sequence consisting of pairwise distinct point λn
and having no limit points in Ω or as a function identically equal to infinity on Ω. With each
nonzero function f≡ 0inH(Ω), the divisor Zfof its zeros which is equal at each point λ∈Ω
to the multiplicity of the root of fat λis naturally associated. If f≡0 in Ω, then, by definition,
Zf(λ)≡∞,λ∈Ω.
For a submodule Iof the module P,wesetZI
def
=min{Zf:f∈I}and call ZIthe divisor
of I.
A space (module) Pis said to be stable (with respect to the division by the binomials z−λ)
if the condition f(z)/(z−λ)∈H(Ω) implies that f(z)/(z−λ)∈Pfor all functions f∈Pand
an arbitrary point λ∈C.
With each divisor Λ on Ω, we associate the submodule I(Λ) def
={f∈P:ZfΛ}of the module
P. It is clear that if Pis a stable module and if I(Λ) ={0}(i.e., I(Λ) is a nonzero submodule),
then ZI(Λ) =Λ.IfΛ=Zffor some function f∈P, then, instead of I(Λ), we also write If.We
also note that, generally speaking, If=¯
I(f), i.e.,Ifis not necessarily a principal submodule.
A submodule Iof the module Pis said to admit a local description or to be abundant ∗[1–3]
if I=I(ZI).
Let Pbe a stable module. Then, by construction, a submodule of the form I(Λ) in Pis
abundant. In this case, ¯
I(f)=Ifif and only if the principal submodule ¯
I(f) is abundant.
Abundant submodules (and ideals) and their properties play an important role in the spectral
theory of operators, in questions of approximation, etc. (For more detail, see the introductions
in [1–3] and surveys [5, 6]).
A submodule Iof the module Pis externally stable at a point λ∈C\Ωiff(z)/(z−λ)∈I
for each function f∈I(see [1–3]), and Iis said to be externally stable if it is externally stable at
∗Some authors also use the terms divisorial,local [6], saturated [15], determined (representable)by zeros,or
synthesized submodule [5], etc.
54
each point λ∈C\Ω. A submodule I⊂Pis internally stable at a point λ∈Ω if the conditions
f∈Iand Zf(λ)>Z
I(λ) imply that f(z)/(z−λ)∈I,andIis sad to be internally stable if it
is internally stable at each point λ∈Ω. A submodule I⊂Pis stable if it is both externally and
internally stable. It is clear that every abundant submodule in a stable module is stable.
AspacePis said to be uniformly stable if, for an arbitrary neighborhood of zero V⊂P, there
is a neighborhood of zero U⊂Psuch that the set {f(z)/(z−λ)∈H(Ω) : f∈U, λ ∈C}is
contained in V.
AspacePis said to be b-stable (see [1–3]) if, for an arbitrary set Bbounded in P, the set
{f(z)/(z−λ)∈H(Ω) : f∈B, λ ∈C}is contained in the space Pand is bounded in it.
A uniformly stable space Pis b-stable. Clearly, a uniformly stable or b-stable space is stable.
Stability conditions require that the topology of Pbe in a sense “soft.” In particular, in view of
these conditions, the Banach spaces of analytic functions with “rigid” topologies prescribing the
existence of more or less explicit boundary values of the functions lie outside the range of application
of the result in this paper. At the same time, a wide class of spaces of holomorphic functions is
already included in the classes of uniformly stable or b-stable spaces (see Proposition 5.1) if there
is even a very slight “gap” between the seminorms determining this topology.
A space (module) Pis said to be analytically densified (see [2, Sec. 5]) if, for an arbitrary finite
system f1,...,f
k∈P, the set {f∈H(Ω) : |f(z)||f1(z)|+···+|fk(z)|,z∈Ω}⊂Pis bounded
in P.
Let ∈P⊂H(Ω). If ≡ 0 and if there is a function m(z), z∈Ω, satisfying the condition
inf
z∈Km(z)>0and sup
z∈K
m(z)<+∞for an arbitrary compact set K⊂Ω(∗)
and possessing the property that the set
{f∈P:|f(z)|max{|(z)|,m(z)},z∈Ω}(1.1)
is bounded in P, then the function is said to be weakly bounding (in P). If the stronger condition
that the set
{f∈H(Ω) : |f(z)|max{|(z)|,m(z)},z∈Ω}(1.2)
is contained in the space Pand is bounded in it holds in this definition (Pis replaced by H(Ω)
in (1.1)), then the function is said to be densely weakly bounding (in P).
The above two definitions impose rather weak constraints on the related objects. For example,
if Pis an analytically densified space and if, for a function ∈P, there is a function g∈Phaving
no common zeros with (in particular, if Pcontains a nonvanishing function g), then, setting
m=||+|g|in (1.2), we conclude that is densely weakly bounding in P.
Let ∈P⊂H(Ω). If ≡ 0 and if there is a function msatisfying condition (∗) and also a
continuous function Φ(z)1, z∈C,increasing faster than any power function, i.e.,
log(1 + |z|)=o(log Φ(z)),z→∞,(1.3)
such that the set
{f∈P:|f(z)|Φ(z)max{|(z)|,m(z)},z∈Ω}(1.4)
is bounded in P, then the function is said to be bounding (in P). If, instead of (1.4), the stronger
condition that the set
{f∈H(Ω) : |f(z)|Φ(z)max{|(z)|,m(z)},z∈Ω}(1.5)
is contained in the space Pand is bounded in it holds in this definition (Pis replaced by H(Ω)
in (1.4)), then the function is said to be densely bounding (in P).
For a bounded domain Ω, every (densely) weakly bounding function in Pis automatically
(densely) bounding with an arbitrary continuous function Φ 1 in (1.5) or in (1.4). In other words,
the appearance of the function Φ in (1.4) or (1.5) is essential only for unbounded domains Ω.
55
Theorem 1. Let P⊂H(Ω) be an analytically densified and sequentially complete module
satisfying one of the conditions [us] or [bs] below.
[us] Pis a uniformly stable space,
[bs] Pis a bornological b-stable space.
Let an abundant submodule Iin Pcontain a bounding function . Then it is generated by two
submodules Iand Ig, where gis a function belonging to I. In particular, if all principal submodules
in Pare abundant∗, then Iis 2-generated. More precisely, I=¯
I(, g)for some function g∈I.
The condition of sequential completeness for the module Pcan be removed here if is densely
bounding.
In what follows, the topological closure of a set Bwill be denoted by B.
The theorem below deals with a more “individualized” version of the same topic as in Theorem 1.
Theorem 2. Let Ωbe either a bounded simply connected domain satisfying the condition
C\Ω=C\Ω (1.6)
or the space Citself, Ω=C,letPbe the same module as in Theorem 1,andletIbe a closed
internally stable submodule in P.IfIcontains an abundant submodule Lwith a bounding function
∈L, then Iis also abundant and is generated by a pair consisting of the submodule Land the
principal submodule ¯
I(g)generated by some function g∈I, i.e., I=L+¯
I(g).
Moreover, if, under the same conditions, Icontains a bounding function generating the abun-
dant principal submodule ¯
I(), then the submodule Iis abundant, and there is a function g∈I
such that I=¯
I(, g).
2. Constructing the Function gfrom a Given Function
Given a function , we suggest a method of constructing the function gfrom with a set
of analytic properties which is needed in the proof of Theorems 1 and 2 to ensure the required
“closeness” between gand .
We denote by D(λ, t)anopendiskofradiustwith center at λ∈C. In particular, if t0,
then D(λ, t) is an empty set.
To represent the divisor Λ ≡∞, supp Λ = {λn},kn=Λ(λn), in Ω we will use the canonical
form Λ={(λn,k
n)}as well.
Proposition 2.1. Let Ω,0∈Ω, be a domain, let ∈H(Ω),andlet(0) = 1. Let a divisor M
be a subdivisor of the divisor of zeros of , i.e., MZ.
Then, for an arbitrary function msatisfying condition (∗)in Sec. 1and for each continuous
function Φ(z)1,z∈C, increasing faster than any power function in the sense of (1.3), there
is a sequence of functions {n}⊂H(Ω) which converges in the topology of uniform convergence to
some function g∈H(Ω) such that
(1) g(0) = 1 and min{Zg,Z
}=M,
(2) the relation
lim
k,p→∞ sup
z∈Ω
|k(z)−p(z)|
max{|(z)|,m(z)}=0 (2.1)
holds,
(3) the inequality
|g(z)|Cmax{(z),m(z)},where z∈Ωand Cis a constant,(2.2)
holds,
(4) there are two polynomial sequences {pn}and {qn}satisfying the condition limn→∞ pn(0) =
limn→∞ qn(0) = 1 and possessing the property that
lim
n→∞ sup
z∈Ω
|pn(z)(z)−qn(z)g(z)|
Φ(z)max{|(z)|,m(z)}=0.(2.3)
∗Such a module Pis said to be pure. (See [2, Sec. 6, Subsec. 1]).
56
Proof. We consider the divisor Λ = Z−M and use its canonical representation Λ = {(λn,k
n)}.
The function gwill be constructed as the limit of the partial sums∗
N(z)=
N
n=1
cn
(z)
(1 −z/λn)kn=(z)
N
n=1
cn
(1 −z/λn)kn,N=1,2,... . (2.4)
Let us concretize the construction of the sequence {cn}. For this, we take a sequence of pairw ise
disjoint disks D(λn,ε
n), εn>0, n=1,2,..., such that
|(z)|inf
ζ∈D(λn,εn)m(ζ),z∈D(λn,ε
n),n=1,2,... . (2.5)
This can always be done consecutively for n=1,2,... according to the maximum principle for ||
andinviewofcondition(∗) on the functions msince (λn)=0.
We now select a sequence{cn}satisfying the following conditions:
(c0)cn>0, n=1,2,...,andn1cn=1,
(c∗) the estimate
aN
nN+1
cn|λ|kn
εkn
n
=o(1) as N→∞ (2.6)
holds, where
aN=
N
n=11+supz∈Ω|z|
|λn|kn
if Ω is bounded,
sup
z∈Ω
N
n=11+ |z|
|λn|kn
Φ(z)if Ω is unbounded.
(2.7)
If the domain Ω is bounded, then the coefficients aN>0 in (2.7) are obviously finite, and if it is
unbounded, then this is true since Φ increases faster than any power function. The required choice
of the sequence {cn}can also be realized on the basis of the elementary lemma below.
Lemma 2.1. Let {an}and {bn},n=1,2,...,be sequences of positive numbers. Then there
is a sequence {cn}such that condition (c0)is fulfilled and the relation aNnN+1 cnbn=o(1) as
N→∞ holds.
Proof. We set s1= 1 and sn=1/an−1>0, n=2,3,.... By construction, the sequence
{s
n},s
1=1,s
n=min{s1,...,s
n}/(n−1), n2, is strictly monotone decreasing. We write
c
n=min{(s
n−s
n+1)/bn,1/n2}>0. By construction, the series n1c
nn11/n2converges
toanumberS>0 in such a way that the relation nN+1 c
nbnnN+1(s
n−s
n+1)=s
N+1
sN+1/N =1/(NaN) holds. Setting cn=c
n/S, we obtain the desired sequence {cn}.
Applying Lemma 2.1 for aNin (2.7) and for bn=|λ|kn/εkn
n, we obtain a sequence {cn}with
properties (c0)and(c∗). We thus derive an upper estimate for the moduli of the terms in (2.4).
For z/∈D(λn,ε
n), we have
(z)
(1 −z/λn)kn
|(z)||λn|kn
εkn
n
,z/∈D(λn,ε
n).(2.8)
For z∈D(λn,ε
n), by the maximum principle in (2.5), it follows that
(z)
(1 −z/λn)kn
m(z)|λn|kn
εkn
n
,z∈D(λn,ε
n).(2.9)
Hence, for all z∈Ω, formulas (2.8) and (2.9) imply that
cn
(z)
(1 −z/λn)kn
max{|(z)|,m(z)}cn
|λn|kn
εkn
n
,z∈Ω,(2.10)
∗As usual, the sum (product) of an empty set of elements is understood as being equal to zero (unity).
57
whence, for 1 pk, by the definition of the functions Nin (2.4), it follows that
|k(z)−p(z)|max{|(z)|,m(z)}
k
n=p+1
cn
|λn|kn
εkn
n
,z∈Ω.(2.11)
By condition (c∗) for the choice of the sequence {cn}, where the sequence {aN}defined by for-
mula (2.7) is nondecreasing, the series on the left-hand side of (2.6) is convergent. Consequently,
(2.11) implies (2.1), and the sequence {N}in (2.4) is uniformly convergent (together with its all
derivatives) on the compact sets to the function
g(z)=
∞
n=1
cn
(z)
(1 −z/λn)kn∈H(Ω) (2.12)
satisfying estimate (2.2). Furthermore, by definition (2.4) of the functions Nandinviewofcondi-
tion (c0), we have g(0) = n1cn= 1. Thus, properties (2) and (3) in the proposition hold, and,
to prove that (1) is fulfilled, it remains to show that min{Zg,Z
}=M.
If λ/∈supp Λ, i.e., Λ(λ) = 0, then each of the terms in (2.12) has a root of multiplicity Z(λ)
(possibly of multiplicity 0) at the point λ. Consequently, the function ghas a root of multiplicity
no lower than Z(λ)atλ, and the relation
min{Z(λ),Z
g(λ)}=Z(λ)=Z(λ)−Λ(λ)=M(λ),λ/∈supp Λ,(2.13)
holds. If λ=λn∈supp Λ, i.e., Λ(λ)>0, then each of the terms in (2.12) except the nth
summand has a root of multiplicity Z(λn) at the point λn, and this nth summand has a root of
multiplicity Z(λn)−kn=Z(λn)−Λ(λn)=M(λn)<Z
(λn)atλn. Thus, the function ghas
a root of multiplicity Zg(λ)=M(λ)<Z
(λ) at each of the points λ∈supp Λ , and the relation
min{Z(λ),Z
g(λ)}=Zg(λ)=M(λ), λ∈supp Λ, holds. This, together with (2.13), implies that
min{Zg,Z
}= M, i.e., property (1) in the proposition holds.
It remains to establish property (4), i.e., relation (2.3). We set
qN(z)=
N
n=1 1−z
λnkn
(2.14)
and, with regard to representations (2.12) and (2.4), consider the identity
qN(z)g(z)=(z)pN(z)+qN(z)RN(z),(2.15)
where
pN(z)=
N
n=1
cnqN(z)
(1 −z/λn)kn,R
N(z)=
∞
n=N+1
cn(z)
(1 −z/λn)kn.(2.16)
Here, by the construction in (2.14), the function pNis a polynomial, and we have
pN(0) =
N
n=1
cn→1asN→∞.(2.17)
According to (2.10), the remainder RNof the series satisfies the inequality
|RN(z)|max{|(z)|,m(z)}
∞
n=N+1
cn
|λn|kn
εkn
n
,N=1,2,... . (2.18)
Furthermore, definition (2.14) implies the inequality
|qN(z)|
N
n=1 1+ |z|
|λn|kn
,q
N(0) = 1,N=1,2,... . (2.19)
58
By estimates (2.18) and (2.19), it follows from representation (2.15)–(2.16) that
|pN(z)(z)−qN(z)g(z)||qN(z)RN(z)|
N
n=1 1+ |z|
|λn|kn
max{|(z)|,m(z)}
∞
n=N+1
cn
|λn|kn
εkn
n
.(2.20)
If Ω is a bounded domain, then we rewrite (2.20) using notation (2.7) and then, by relation (2.6),
derive (2.3) from condition (c∗) on the sequence{cn}since Φ(z)1. If Ω is an unbounded do-
main, then, preliminarily dividing and multiplying the right-hand side of (2.20) by Φ(z), we again
derive (2.3) from (2.20) using the same procedure as above. Thus, the proof is complete.
Remark. Along with the additive method of constructing the function gas the limit of the
partial sums Nin (2.4), an alternative multiplicative method is also possible (cf. [11–13, 16]) in
which gis constructed as the limit of a recurrent sequence 0=,n(z)=(1−z/γn)knn−1(z)/(1−
z/λn)kn∈H(Ω), n=1,2,..., where {γn}is sufficiently close to the sequence {λn}, but has no
common points with it. This method leads to the same result, but, in our realization, it turned out
to be much more extensive and laborious.
Proposition 2.2. Let 0∈Ω,letP⊂H(Ω) be a stable sequentially complete space, and let
∈Pbe a weakly bounding function. Then the function gconstructed as in Proposition 2.1 belongs
to P. The conditions of sequential completeness and stability for the space Pcan be removed here
if is a densely weakly bounding function.
Proof. If gis a densely weakly bounding function in P, then, according to estimate (2.2) for g,
it belongs to the set (1.2), which is contained in P. Hence, g∈P, which proves the second part
of the propositions.
If is only a weakly bounding function belonging to the stable sequentially complete space P,
then, by the stability of this space, all the function Nconstructed by rule (2.4) belong to P.Since
the set (1.1) is bounded, relation (2.1) in Proposition 2.1 means that {n}is a Cauchy sequence.
Consequently, its limit in P, which coincides with gsince the topology of the space Pmajorizes
the topology of pointwise convergence, also belongs to P.
3. On Submodules Generated by Two Submodules
Throughout this section, Ω is a domain in C,0∈Ω, and it is assumed that the module P
satisfies either condition [us] or condition [bs]. (See Theorem 1.) By Remark 2 in [2, Sec. 4, Sub-
sec. 2], the condition of pointwise stability holds in the case [bs] and, obviously, in the case [us] as
well. Thus, the assertion below is true.
Let Vbe an arbitrary neighborhood of zero in P. Then, for each λ∈C, there is a neighborhood
Uλsuch that the conditions f∈Uλand f(z)/(z−λ)∈H(Ω) imply that f(z)/(z−λ)∈V.
Preliminarily, we present a list of some more results in [2] needed in what follows and give some
comments when necessary.
(i) If a net (generalized sequence){fσ}⊂Pover a directed set {σ}converges in Pto an
element f∈P, then, for each n=0,1,...,thenth derivatives satisfy the condition f(n)
σ(λ)→
σ
f(n)(λ)at all points λ∈Ω. (See [2, Proposition 4.5 and Sec. 4, Subsec. 2, Remark 2].) This property
ensures the preservation of the divisor of a submodule under the transition to its closure, and we
will use it in what follows without any additional reference.
(ii) A closed submodule in Pinternally stable at a point in Ωis internally stable. (See [2,
Proposition 4.2 and Sec. 4, Remark 1].)
(iii) The closure ¯
Iof a submodule I⊂Pstable at a point λis stable at the same point (see [2,
Proposition 4.5 and Sec. 4, Subsec. 2, Remark 2]).
(iv) In an analytically densified module P, every closed stable submodule containing an abundant
nonzero submodule is also abundant [2] (see Proposition 6.4). This result was stated in [2] for
submodules over C[z] of local rank 1 in modules of vector functions. But in the case of scalar
59
functions, this is just as well true without the above additional assumption since every subset in
H(Ω) distinct from {0}is of local rank 1. (See [2, Sec. 1, Subsec. 3].) Moreover, this result was
stated in [2] only for the case [us] without any comment on the case [bs]. By the assertion in the
very end of [2, Sec. 5], it is readily seen that the indicated result is true in the case [bs] as well.
The two propositions below are a slightly different version of essentially more general assertions
in [2] (namely, of Propositions 4.8 and 6.6). Cf. [17, Proposition 5.6].
Proposition 3.1. Let Jbe a closed submodule in P⊂H(Ω) generated by a pair of submodules
Land G,let∈Land g∈G,(0) = g(0) = 1, and let the following assumption hold:
[ss] there are two polynomial sequences {pn},{qn}∈C[z],n=1,2,..., satisfying the condition
lim
n→∞ pn(0) = lim
n→∞ qn(0) = 1 (3.1)
and possessing the property that the sequence {pn−qng},n=1,2,..., tends to 0∈Pin the
topology of the spacePas n→∞.
If the submodules Land Gare internally stable at the point 0∈Ω, then the submodule Jis
also internally stable.
Proof. We first somewhat modify the sequences {pn}and {qn},n=1,2,... . By the definition
of a topological vector space, it follows from condition (3.1) that (pn(0) −1)−(qn(0) −1)g→
n0
in P. What has been established implies that the sequence {(pn−(pn(0)−1))−(qn−(qn(0)−1))g}
converges to 0 in P, and the polynomials for and gare already equal to 1 at zero. Consequently,
condition (3.1) in the proposition can be replaced by
pn(0) = qn(0) = (0) = g(0) = 1,n=1,2,..., p
n−qng→
n0inP.(3.2)
It is clear that ZJ(0) = 0 since (0) = 1 = 0 and ∈L⊂J. We consider an arbitrary element
F∈Jsuch that ZF(0) >Z
J(0) = 0. By (ii), it suffices to show that F(z)/z ∈J.
Let Vbe a neighborhood of zero in P. It follows from the remark at the beginning of this
section that, in each of the cases [us] and [bs], the module Pis pointwise stable. Hence, there is a
neighborhood of zero U0in Psuch that the implication
f∈U0,f(0) = 0 =⇒f(z)
z∈V(3.3)
is true. We now choose a neighborhood of zero Wsuch that W+W⊂U0.SinceF∈J, there is
anetFσ=σ−gσ∈L+Gover the directed set {σ},σ∈L,gσ∈G, convergent to Fin P.
We modify it in such a way that the terms of the net vanish at the point 0 ∈Ω. For this, it
suffices to consider another net F∗
σ=Fσ−(σ(0) −gσ(0))=σ−gσ−(σ(0) −gσ(0))∈L+G,
which, as before, converges to Fin Psince Fσ(0) →F(0) = 0, (0) = 1, and, accordingly,
(σ(0)−gσ(0))→0inP. Here, the new net contains a term having the form F∗=fL−fG∈L+G,
where fL=σ−(σ(0)−gσ(0))∈L,fG=gσ∈G,andF∗(0) = fL(0) −fG(0) = 0, and possessing
the property that F−F∗∈W.Letussethn=fL(0)(pn−qng)=fL(0)pn−fG(0)qng∈Wfor a
sufficiently large n. For this value of n,wehaveF−F∗+hn∈W+W⊂U0, and, in view of (3.3),
the division by znow gives
F(z)
z−F∗(z)−hn(z)
z=F(z)
z−fL−fL(0)pn
z−fG−fG(0)qng
z∈V,
where, with regard to (3.2) and by the internal stability of Land Gat 0 ∈Ω, the inclusion relations
(fL−fL(0)pn)/z ∈Land (fG−fG(0)qng)/z ∈Ghold.
By the arbitrariness in the choice of V, these relations mean that F(z)/z is a point of contact
for the sum L+G, i.e., F(z)/z ∈J, which is what we had to prove.
Proposition 3.2. Let, in addition to property [us] or [bs], the module Pbe analytically densi-
fied and let the assumptions of Proposition 3.1 including condition [ss] also hold. If the submodules
Land Gare abundant, then the submodule J=L+Ggenerated by them is also abundant.
60
Proof. As was noted in Sec. 1, by the definition of the property of abundance, each of the
submodules Land Gis stable. By definition, Jis a closed submodule. According to Proposition 3.1,
it is internally stable. Since each of the submodules Land Gis externally stable, their sum L+G
is also externally stable. (This sum is regarded here without closure. See the definition of external
stability.) In the topological module, the closure J=L+Gof the submodule L+Gis also a
submodule, and, according to (iii), the submodule Jis also externally stable. Thus, Jis closed and
both internally and externally stable, i.e., it is a stable submodule. By construction, Jcontains
the abundant submodule L(or G), whence, since the module Pis analytically densified, it follows
that, according to (iv), the submodule Jis abundant, which is what is required to prove.
4. Proofs of Theorems 1 and 2
Using the shift of the complex plane, it can always be assumed without loss of generality in
the proofs of Theorems 1 and 2 that 0 ∈Ωand(0) = 1. Throughout this section, it will also
be assumed that the module Psatisfies one of the conditions [us] and [bs] in the statement of
Theorem 1.
The two facts presented below continue the list of the results in [2] (see the previous section)
we need.
(v) A principal submodule in Pis always internally stable. (See [2, Sec. 4, Susbec. 3, Corol-
lary 2].) This assertion was proved only in the case [us], but an analysis of the proof of Proposi-
tion 4.7 in [2], whose corollary has just been stated, shows that only the property of b-stability and
the property of pointwise stability, which follows from [bs], are in fact used in this proposition.
(vi) For a bounded simply connected domain Ωsatisfying condition (1.6), every closed submod-
ule I⊂Pis externally stable. (See [2, Proposition 4.4 and Sec. 4, Susbec. 1, Remark 1].)
Proposition 4.1. Let Pbe the same module as in Theorem 1,letbe a bounding function
in P,andletMbe a divisor such that MZ. Then there is a function g∈Psuch that the
submodule Jgenerated by the submodules Iand Igis abundant and the relation ZJ=Mholds.
The condition of sequential completeness for the module Pcan be removed if is a densely bounding
function.
Proof. We cho o s e gin the form of a function constructed in Proposition 2.1. Proposition 2.2
implies that g∈P.
Both the submodules Iand Igare abundant since they have the form I(Λ). (See Sec. 1.) By
the boundedness of the set (1.4) (of the set (1.5) if is densely bounding), relation (2.3) in property
(4) stated in Proposition 2.1 means that condition [ss] in Proposition 3.1 holds. By Proposition 3.2,
it follows that the submodule J=I+Igis abundant. In this case, by the construction of the
submodules J,I,andIg,wehaveZJ=min{Z,Z
g}. According to property (1) in Proposition 2.1,
the relation min{Z,Z
g}= M holds, and hence ZJ= M, which proves the proposition.
Proof of Theorem 1. Let M = ZI. It is clear that M Z. By Proposition 4.1, the submodule
J=I+Igis abundant as well as the submodule I, and we have ZJ=M=ZI. Two abundant
submodules with the same divisors coincide, i.e., I=J. If all principal submodules in Pare
abundant, then (see Sec. 1) I=¯
I()andIg=¯
I(), i.e., J=¯
I()+¯
I(g), which proves the
theorem
Proposition 4.2. Let the module P, the domain Ω, and the function be respectively the same
as in Theorem 1, Theorem 2, and Proposition 4.1.LetLbe an abundant submodule in Pwhich
contains and let Mbe a divisor such that MZL. Then there is a function g∈Psuch that the
submodule Jgenerated by the submodule Land by the principal submodule ¯
I(g)is abundant and
the relation ZJ=Mholds.
Proof. The condition M ZLimplies that M Z. We again choose gin the form of a
function constructed in Proposition 2.1. By Proposition 2.2, it belongs to P. According to (v), the
principal submodule ¯
I(g) is internally stable. The submodule Lis stable since it is abundant. Since
the set (1.4) (the set (1.5) if is densely bounding) is bounded, relation (2.3) in assertion (4) of
61
Proposition 2.1 means that condition [ss] in Proposition 3.1 holds. Consequently, by Proposition 3.1,
the submodule Jgenerated by Land ¯
I(g) is internally stable. It follows from (vi) that the closed
submodule Jis externally stable if the domain Ω satisfies condition (1.6), i.e., Jis stable in this
case. Since Jcontains the abundant submodule L, it follows that, according to (iv), Jis abundant.
Furthermore, by property (1) of the function gin Proposition 2.1, we have min{Z,Z
g}=M,
whence
ZJ=min{ZL,Z
g}min{Z,Z
g}=M.(4.4)
By the condition M ZLand according to the construction, we have M Zg, i.e., M
min{ZL,Z
g}, which, together with (4.4), gives M = ZJ, and the proposition is thus proved.
Proof of Theorem 2. We take the divisor M = ZI. The inclusion relation L⊂Iimplies that
MZL. The submodule J=L+Igin Proposition 4.2 is abundant.
By assumption, the submodule Iis closed and internally stable, and, according to (vi), it is
externally stable as well. Therefore, Iis a closed stable submodule. Here Iincludes an abundant
submodule L, whence, by (iv), it follows that Iis also an abundant submodule. Moreover, we
have ZJ=M=ZI. Consequently, the two abundant submodules Iand Jcoincide, which proves
the first part of the theorem.
To deduce the second part from the above, it suffices to note that the abundant principal
submodule ¯
I() can be taken as the abundant submodule L⊂Ientering the assumptions of the
theorem.
5. Applications
We first single out a rather wide class of spaces to which Theorems 1 and 2 are applicable.
Let qbe a real-valued function in Ω. We denote by Pqthe normed space consisting of all
functions f∈H(Ω) with finite norm fq=sup
ζ∈Ω(|f(ζ)|exp(−q(ζ))).
Let Q={qn},n∈N, denote an increasing (decreasing) sequence of real-valued functions on Ω,
and let us introduce the locally convex space P↑
Q=n1Pqn(the space P↓
Q=n1Pqn)withthe
natural topology of inductive (projective) limit. (See [4, 18].) We also introduce the function lΩ(z)=
max{1+|z|,1/dist(z,∂Ω)},z∈Ω , where dist( ·,·) is the distance function and ∂Ω is the boundary
of Ω.
Recall (see [18]) that the inductive limit P↑
Q(projective limit P↓
Q)isan(LN)∗-space
((M)∗-space) on condition that, for each index n, there is an index k>n(for each kthere is an
index n>k) such that the embedding of Pqnin Pqkis completely continuous, i.e., it transforms
a ball in the space Pqninto a relatively compact subset in Pqk. The (LN )∗-spaces ((M)∗-spaces)
possess some useful properties essentially facilitating the operation with them. (See [18].)
Proposition 5.1 (see [16, Theorem 2]).Let an increasing (decreasig)sequence Q={qn}of
continuous functions in Csatisfy the following conditions:
[Qs] for an arbitrary index n(m), there are indices mand ksatisfying the inequalities mkn
(indices nand ksatisfying the inequalities nkm)and a constant C0such that, for each
point λ∈Ω, the inequality supζ∈Dqk(ζ)infζ∈Dqm(ζ)+Cholds in the domain
D=Dn,k(λ)=ζ∈Ω:|ζ−λ|<exp(qn(ζ)−qk(ζ))
3lΩ(ζ),
[Ql] for an arbitrary index m(p), there is an index pm(mp)and a constant C0
such that qm(z)+loglΩ(z)qp(z)+C,z∈Ω.
Then the inductive (projective)limit P↑
Q(P↓
Q)is a bornological b-stable (uniformly stable)
separable complete reflexive analytically densified (LN )∗-space ((M)∗-space)and, simultaneously,
a topological module over the ring C[z].
For a 2 π-periodic ρ-trigonometrically convex (see[19]) lower semicontinuous function H(θ)>0,
θ∈R, the module of entire functions [ρ, H )forρ>0([ρ, H]forρ>1) with indicator, for the
order ρ, strictly less (no less) than H(θ), θ∈[0,2π), is a space of the form P↑
Q(P↓
Q) dealt with
62
in Proposition 5.1. Consequently, Theorem 1 and 2 can be used in these spaces. Following this
approach and using the well-known duality scheme when necessary, it is easy to obtain some of
the main results in [11, Theorems 1 and 2], [12, Theorem 3], and [13, Theorems 3.6, 4.6, and 4.7]
on 2-generated submodules and on the representability of differentiation-invariant subspaces in the
form of a solution space of two homogeneous convolution equations. To demonstrate another appli-
cation of the main results to spectral synthesis, we resort to the classical space C∞(a, b) of infinitely
differentiable complex-valued functions on an open (bounded or unbounded) interval (a, b)⊂R,
−∞ a<b+∞. (In what follows, some facts given in [4,20, 21] are used.)
Let Kn=[an,b
n]⊂[an+1,b
n+1], n=0,1,..., be closed intervals in Rand let their union
coincide with (a, b). We denote by Cn(Kn) the space of functions continuously differentiable up to
the order non [an,b
n] with the norm fn=sup
x∈Knmax0kn|f(k)(x)|,f∈Cn(Kn). The natural
topology on the space C∞(a, b)=∞
n=0 Cn(Kn) is introduced as the projective-limit topology of
the space Cn(Kn).
Following L. Schwartz, we denote the strong dual of C∞(a, b), which is the space of distribu-
tions compactly supported in the interval (a, b), by E
(a, b) since Schwartz originally denoted the
space C∞(a, b) by the symbol E(a, b). The space ˆ
E
(a, b) consists of the Fourier–Laplace trans-
forms ˆ
S(z)=S, exp(−izx),z∈C, of all distributions S∈E
(a, b), and its topology is induced
by that of E
(a, b) under the Fourier–Laplace transformation. The correspondence determined by
this transformation is one-to-one. For J⊂E
(a, b), we set ˆ
J={ˆ
S:S∈J}.Ifl∈ˆ
E
(a, b), then ˇ
l
will denote the compactly supported distribution whose Fourier–Laplace transform is l.
Furthermore, we introduce the spaces
Pn=l∈H(C):l
n=sup
z∈C
|l(z)|
(1 + |z|)nexp(bnIm+z−anIm−z)<∞,
where Im±(z)=max{0,±Im z}. By one of the Paley–Wiener-Schwartz theorems, the space ˆ
E
(a, b)
can be internally described as the inductive limit of the normed spaces Pn,n=1,2,..., with the
norms ·
n.
It can be seen easily that ˆ
E
(a, b) is a space of the form P↑
Qsatisfying the assumptions of
Proposition 5.1 and corresponding assertion of the proposition holds for it.
Theorem 3. Every abundant submodule Iin the module ˆ
E
(a, b)is generated by a pair of
submodules of the form Iand Igfor some , g ∈I.
Proof. By Theorem 1, it suffices to show that Icontains a bounding function . Without loss
of generality, we can consider the space ˆ
E
(−a, a)witha>0.
First, let f∈I. Then there are numbers k∈N,a,anda,0<a
<a, such that |f(z)|
C(1+|z|)kexp(a|Im z|), z∈C, where Cis a constant. We take a number a >0 such that
a<a
<aand an exponential-type entire function h≡ 0 satisfying, for instance, an estimate
of the form |h(z)|exp(ε|Im z|−|z|), where 0 <ε<a−a . (Such a function always exists.
For example, see [22, Theorems 39 and 40].) In view of the estimates for fand h, the product
=fh belongs to Pn⊂ˆ
E
(−a, a) for some nksuch that a +ε<a
n. Moreover, since Iis
abundant, we have ∈Iin view of the relation ZIZfZ.Ifm(z)≡1andΦ(z) = exp |z|,
then the set (1.5) is contained in the space Pnand is bounded in it. For ˆ
E
(−a, a) regarded as
an (LN)∗-space, this means (see [18, Theorem 2]) that the set (1.5) is bounded in ˆ
E
(−a, a). Hence,
is a bounding function in I, which is what we had to prove.
For S∈E
(a, b) on the interval {h∈R: co supp S+h⊂(a, b)}, where co supp Sis the convex
hull of the set supp S, the convolution (S∗f)(h)=S, f(x+h)is defined for all f∈C∞(a, b).
The (differentiation) invariant subspace W⊂Cn(a, b)admits spectral synthesis if it coincides
with the closure of the linear span of the functions of the form xmeiλx ∈Win the variable x∈(a, b).
The assertion below is established following the standard scheme. (Compare with [21, Sec. 4] and
[24, Sec. 2].)
63
Duality principle. There is a one-to-one correspondence between the set of all closed invariant
subspaces W⊂C∞(a, b)and the set of all closed submodules I⊂ˆ
E
(a, b)which is established
according to the following orthogonality rule: W↔Iif and only if I=ˆ
J, where the functionals
in J⊂E
(a, b)and only they annihilate W. An invariant subspace Wadmits spectral synthesis if
and only if the corresponding closed submodule I⊂ˆ
E
(a, b)is abundant.
Following [17, Sec. 6], we refer to the closed submodule I∈ˆ
E
(a, b) corresponding to a closed
invariant subspace W∈C∞(a, b) in accordance with the duality principle as the annihilator sub-
module for W. The annihilator submodule for the intersection n
k=1 Wkcoincides with the closed
submodule generated by the annihilator submodules I1,...,I
nfor the corresponding closed invari-
ant subspaces W1,...,W
n.
Recall that the solution space WSof a single homogeneous convolution equation S∗f≡0
in E(a, b), where S∈E
(a, b), is invariant, and it admits spectral synthesis. (See [23, Theorem 2],
[20, Theorem 16.4.1], and [21, Sec. 20].) By the duality principle, this means that the annihilator
submodule for WSis exactly Iˆ
S={f∈ˆ
E
(a, b):ZfZˆ
S}. Thus, if a submodule Iis generated
by a pair of submodules of the form Iand Ig,, g ∈I, then, for a closed invariant subspace W
with annihilator submodule I, this implies the representability of Win the form of the intersection
of two invariant subspaces with annihilator submodules Iand Ig, i.e., Wis the solution space for
the system of two homogeneous convolutions equations ˇ
∗f=ˇg∗f= 0. Hence, Theorem 3 can be
restated in the dual form by analogy with what is done in [17, Sec. 6] and [11, Sec. 1], as follows.
Theorem 4. If an invariant subspace in E(a, b)admits spectral synthesis, then it coincides
with the solution space of a system of at most two homogeneous convolution equations.
Remark. Proposition 5.1 can be extended without any difficulties to inductive (projective)
limits of normed spaces of holomorphic functions determined by integrated norms.
Analogs of Theorem 3 can be established for the space of exponential-type entire functions
which are Fourier–Laplace transforms of Beurling or Roumier type ultradisributions (see [25]), for
the space of hyperfunctions (see [20, Chap. 9]) with supports on subintervals compactly embedded in
a fixed interval, for the space of analytic functionals in spaces of holomorphic functions on a convex
domain D⊂Cwith constraints on their growth near the boundary ∂D (see [26]), etc. It seems
that such analogs can be used to obtain results similar to Theorem 4 in terms of convolution type
equations if the convolution is possible or in terms of infinite-order differential equations (see [21,
Secs. 21 and 22]) in spaces of Beurling or Roumier type (see [25]) ultradifferentiable functions, in
spaces of real-analytic function on an interval, and in spaces of holomorphic functions in a convex
domain D⊂Cwith a prescribed rate of growth near the boundary ∂D [26].
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Bashkir State University,
Inst. Math. with Comp. Center, Edu.–Sci. Center, Russian Acad. Sci.
e-mail: algeom@bsu.bushedu.ru, khabib-bulat@mail.ru
Translated by V. M. Volosov
