Noncoherence of a Causal Wiener Algebra Used in Control Theory

Abstract

Let ℂ_≥0∶={𝑠∈ℂ∣Re(𝑠)≥0}, and let 𝒲⁺ denote the ring of all functions 𝑓∶ℂ_≥0→ℂ such that 𝑓(𝑠)=𝑓_𝑎(𝑠)+∑^∞_𝑘=0𝑓_𝑘𝑒^{−𝑠𝑡𝑘}(𝑠∈ℂ_≥0), where 𝑓_𝑎∈𝐿¹(0,∞),(𝑓_𝑘)_𝑘≥0∈ℓ¹, and 0=𝑡₀<𝑡₁<𝑡₂<⋯ equipped with pointwise operations. (Here ^0𝑥0⋅^005𝑒 denotes the Laplace transform.) It is shown that the ring 𝒲⁺ is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain 𝒲⁺ whose intersection is not finitely generated.

1. Introduction

The aim of this paper is to show that the ring 𝒲⁺ (defined below) is not coherent.

We first recall the notion of a coherent ring.

Definition 1.1. Let 𝑅 be a commutative ring with identity element 1, and let 𝑅^𝑚=𝑅×⋯×𝑅 (𝑚 times). Suppose that 𝑓=(𝑓₁,…,𝑓_𝑚)∈𝑅^𝑚.
(1) An element (𝑔₁,…,𝑔_𝑚)∈𝑅^𝑚 is called a relation on   𝑓 if 𝑔₁𝑓₁+⋯+𝑔_𝑚𝑓_𝑚=0.(1.1)(2) Let 𝑓^⟂ denote the set of all relations on 𝑓∈𝑅^𝑚. (Then 𝑓^⟂ is an 𝑅-submodule of the 𝑅-module 𝑅^𝑚.)(3) The ring 𝑅 is called coherent if for all 𝑚∈ℕ and all 𝑓∈𝑅^𝑚, 𝑓^⟂ is finitely generated, that is, there exists a 𝑑∈ℕ and there exist 𝑔_𝑗∈𝑓^⟂, 𝑗∈{1,…,𝑑}, such that for all 𝑔∈𝑓^⟂, there exist 𝑟_𝑗∈𝑅, 𝑗∈{1,…,𝑑} such that 𝑔=𝑟₁𝑔₁+⋯+𝑟_𝑑𝑔_𝑑.

An integral domain is coherent if and only if the intersection of any two finitely generated ideals of the ring is again finitely generated; see [1, Theorem 2.3.2, page 45].

The coherence of some rings of analytic functions has been investigated in earlier works. For example, McVoy and Rubel [2] showed that the Hardy algebra 𝐻^∞(𝔻) is coherent, while the disc algebra 𝐴(𝔻) is not. Mortini and von Renteln proved that the Wiener algebra 𝑊⁺(𝔻) (of all absolutely convergent Taylor series in the open unit disc) is not coherent [3]. In this article, we will show that the ring 𝒲⁺ (defined below, and which is useful in control theory) is not coherent.

Notation 1. Throughout the article, we will use the following notation:
ℂ_≥0∶={𝑠∈ℂ∣Re(𝑠)≥0}.(1.2)

Definition 1.2. Let 𝒲⁺ denote the Banach algebra𝒲⁺={𝑓∶ℂ_≥0→ℂ|𝑓(𝑠)=𝑓_𝑎(𝑠)+∑^∞_𝑘=0𝑓_𝑘𝑒^{−𝑠𝑡𝑘}(𝑠∈ℂ_≥0),𝑓_𝑎∶(0,∞)→ℂ,𝑓_𝑎∈𝐿¹(0,∞),∀𝑘≥0,𝑓_𝑘∈ℂ,(𝑓_𝑘)_𝑘≥0∈ℓ¹,∀𝑘≥0,𝑡_𝑘∈ℝ,0=𝑡₀<𝑡₁<𝑡₂<⋯}(1.3)equipped with pointwise operations and the norm‖𝑓‖_𝒲+∶=‖𝑓_𝑎‖_𝐿1+‖(𝑓_𝑘)_𝑘≥0‖_ℓ1.(1.4)Here 𝑓_𝑎 denotes the Laplace transform of 𝑓_𝑎, given by𝑓_𝑎(𝑠)=^∞₀𝑒^−𝑠𝑡𝑓_𝑎(𝑡)𝑑𝑡,𝑠∈ℂ_≥0.(1.5)

The above algebra arises as a natural class of transfer functions of stable distributed parameter systems in control theory; see [4, 5].

Our main result is the following.

Theorem 1.3. The ring 𝒲⁺ is not coherent.

The relevance of the coherence property in control theory can be found in [6, 7]. We will prove Theorem 1.3 following the same method as in the proof of the noncoherence of 𝑊⁺(𝔻) given by Mortini and von Renteln in [3].

In Section 3, we will give the proof of Theorem 1.3. But before doing that, in Section 2, we first prove a few technical results needed in the sequel.

2. Preliminaries

We first recall the definition of the Hardy algebra 𝐻^∞ of the open right half plane.

Definition 2.1. Let 𝐻^∞ denote the Hardy space of all bounded analytic functions in the open right half plane equipped with the norm‖𝜑‖_∞∶=sup_Re(𝑠)>0|𝜑(𝑠)|,𝜑∈𝐻^∞.(2.1)

In order to prove our main result (Theorem 1.3), we will use the relation between the convergence in 𝐻^∞ versus that in 𝒲⁺.

Lemma 2.2. If 𝑓∈𝒲⁺, then 𝑓∈𝐻^∞ and ‖𝑓‖_∞≤‖𝑓‖_𝒲+.

Proof. Let𝑓(𝑠)=𝑓_𝑎(𝑠)+^∞_𝑘=0𝑓_𝑘𝑒^{−𝑠𝑡𝑘}(𝑠∈ℂ_≥0).(2.2)For 𝑠∈ℂ_≥0, we have|𝑓_𝑎(𝑠)|=|^∞₀𝑒^−𝑠𝑡𝑓_𝑎(𝑡)𝑑𝑡|≤^∞₀𝑒^{−Re(𝑠)𝑡}|𝑓_𝑎(𝑡)|𝑑𝑡≤^∞₀1⋅|𝑓_𝑎(𝑡)|𝑑𝑡=‖𝑓_𝑎‖_𝐿1,(2.3)and moreover,|^∞_𝑘=0𝑓_𝑘𝑒^{−𝑠𝑡𝑘}|≤^∞_𝑘=0|𝑓_𝑘|𝑒^{−Re(𝑠)𝑡𝑘}≤^∞_𝑘=0|𝑓_𝑘|⋅1=‖(𝑓_𝑘)_𝑘‖_ℓ1.(2.4)So the result follows.

The maximal ideal 𝔪₀ (defined below) of 𝒲⁺ will play an important role in the remainder of this article.

Notation. Let 𝔪₀ denote the kernel of the complex algebra homomorphism 𝑓↦𝑓(0)∶𝒲⁺→ℂ, that is,
𝔪₀∶={𝑓∈𝒲⁺∣𝑓(0)=0}.(1)

Then 𝔪₀ is a maximal ideal of 𝒲⁺, and this maximal ideal plays an important role in the proof of our main result in the next section. We will prove a few technical results about 𝔪₀ in this section, which will be used in the sequel. The following result is analogous to [3, Lemma 1].

Lemma 2.3. Let 𝐿≠(0) be an ideal in 𝒲⁺ contained in the maximal ideal 𝔪₀. If 𝐿=𝐿𝔪₀, that is, if every function 𝑓∈𝐿 can be factorized in a product 𝑓=ℎ𝑔 of two functions ℎ∈𝐿 and 𝑔∈𝔪₀, then 𝐿 cannot be finitely generated.

Proof. Suppose that𝐿=(𝑓₁,…,𝑓_𝑁)≠(0)(2.5)is a finitely generated ideal in 𝒲⁺ contained in the maximal ideal 𝔪₀. By our assumption, there are functions ℎ_𝑛∈𝐿, 𝑔_𝑛∈𝔪₀ with𝑓_𝑛=ℎ_𝑛𝑔_𝑛(𝑛=1,…,𝑁).(2.6)Since ℎ_𝑛∈𝐿, there exist functions 𝑞^(𝑛)_𝑘∈𝒲⁺ withℎ_𝑛=^𝑁_𝑘=1𝑞^(𝑛)_𝑘𝑓_𝑘(𝑛=1,…,𝑁;𝑘=1,…,𝑁).(2.7)From this it follows that^𝑁_𝑛=1|ℎ_𝑛|≤𝑁𝐶^𝑁_𝑛=1|𝑓_𝑛|=𝑁𝐶^𝑁_𝑛=1|ℎ_𝑛𝑔_𝑛|inℂ_≥0,(2.8)where 𝐶 is a constant chosen so that‖𝑞^(𝑛)_𝑘‖_∞≤𝐶,∀𝑘and𝑛.(2.9)(Here ‖⋅‖_∞ denotes the supnorm over ℂ_≥0.) This implies together with the Cauchy-Schwarz inequality that^𝑁∑_𝑛=1|ℎ_𝑛|²≤(^𝑁_𝑛=1|ℎ_𝑛|)²≤𝑁²𝐶²(^𝑁_𝑛=1|ℎ_𝑛𝑔_𝑛|)²≤𝑁²𝐶²(^𝑁_𝑛=1|ℎ_𝑛|²)(^𝑁_𝑛=1|𝑔_𝑛|²).(2.10)This inequality holds for all 𝑠∈ℂ_≥0. With 𝛿∶=1/(𝑁²𝐶²), we obtain the inequality𝛿≤^𝑁_𝑛=1|𝑔_𝑛(𝑠)|²(2.11)for all points 𝑠∈𝐸, where𝐸∶={𝑠∈ℂ_≥0|^𝑁_𝑛=1|ℎ_𝑛(𝑠)|²>0}.(2.12)Since 𝐿≠(0), 𝐸 is a dense subset of ℂ_≥0 (for otherwise, if 𝑠₀∈ℂ_≥0 is such that it has a neighbourhood 𝑉 in ℂ_≥0 where there is no point of 𝐸, then each ℎ_𝑛 is identically zero in 𝑉, and by the identity theorem for holomorphic functions, each ℎ_𝑛 is zero; consequently each 𝑓_𝑛 is zero, and so 𝐿=(0), a contradiction). So by continuity, inequality (2.11) holds in ℂ_≥0. But this contradicts the fact that each 𝑔_𝑛 vanishes at 0.

Remark 2.4. Lemma 2.3 can be proved purely algebraically using Nakayama's lemma. Indeed, it holds in the following more general algebraic situation: if 𝐼 is a nonzero ideal of a commutative domain 𝐷 contained in a maximal ideal 𝑀 and 𝐼=𝐼𝑀, then 𝐼 cannot be finitely generated. However, we have given an analytic proof in our special case above.

Since every maximal ideal is closed, 𝔪₀ is a commutative Banach subalgebra of 𝒲⁺, but obviously without identity element. But there is a substitute, namely the notion of the approximate identity, which turns out to be useful.

Definition 2.5. Let 𝑅 be a commutative Banach algebra (without identity element). One says that 𝑅 has an approximate identity if there exists a bounded sequence (𝑒_𝑛)_𝑛 of elements 𝑒_𝑛 in 𝑅 such that for any 𝑓∈𝑅,lim_𝑛→∞‖𝑒_𝑛𝑓−𝑓‖=0.(2.13)

We will now prove the following result, which shows that the maximal ideal 𝔪₀ in 𝒲⁺ has an approximate identity.

Theorem 2.6. Let 𝑒_𝑛∶=𝑠𝑠+1/𝑛,𝑛∈ℕ.(2.14)Then (𝑒_𝑛)_𝑛∈ℕ is an approximate identity for 𝔪₀.

The existence of an approximate identity for the maximal ideal 𝔪₀ in 𝒲⁺ is not obvious. In order to prove Theorem 2.6, we will need the following lemma.

Lemma 2.7. Suppose 𝑓̂‌∈𝔪₀. Then, for all 𝜖>0, there exists a 𝑝̂∈𝔪₀ such that 𝑝 has compact support in [0,∞), and ‖𝑓̂‌−𝑝‖̂_𝒲+<𝜖.

Proof. Let 𝜖>0 be given. Suppose that𝑓=𝑓_𝑎+^∞_𝑘=0𝑓_𝑘𝛿(⋅−𝑡_𝑘),(2.15)where 𝑓_𝑎∈𝐿¹(0,∞), (𝑓_𝑘)_𝑘≥0∈ℓ¹, and 0=𝑡₀<𝑡₁<𝑡₂<⋯. Since ∫^∞₀|𝑓_𝑎(𝑡)|𝑑𝑡<∞, we can choose an 𝑀>0 large enough such that^∞_𝑀|𝑓_𝑎(𝑡)|𝑑𝑡<𝜖4.(2.16)With 𝑝_𝑎(𝑡)∶=𝑓_𝑎(𝑡) if 𝑡∈[0,𝑀], and 0 otherwise, we have that 𝑝_𝑎∈𝐿¹(0,∞) is compactly supported and‖𝑝_𝑎−𝑓_𝑎‖_𝐿1<𝜖4.(2.17)Furthermore, select 𝑁∈ℕ such that_𝑘_>𝑁|𝑓_𝑘|<𝜖4.(2.18)Now let 𝑇∈(0,∞) be any number satisfying 𝑡_𝑁<𝑇<𝑡_𝑁+1, and define𝑓_𝑇∶=−(^∞₀𝑝_𝑎(𝑡)𝑑𝑡+_{0≤𝑘≤𝑁}𝑓_𝑘).(2.19)Set𝑝∶=𝑝_𝑎+_{0≤𝑘≤𝑁}𝑓_𝑘𝛿(⋅−𝑡_𝑘)+𝑓_𝑇𝛿(⋅−𝑇).(2.20)Then 𝑝̂∈𝒲⁺ and𝑝(̂0)=^∞₀𝑝(𝑡)𝑑𝑡=^∞₀𝑝_𝑎(𝑡)𝑑𝑡+_{0≤𝑘≤𝑁}𝑓_𝑘+𝑓_𝑇=0.(2.21)So 𝑝̂∈𝔪₀. Clearly 𝑝 has compact support contained in [0,∞). We have|𝑓_𝑇|=|^∞₀𝑝_𝑎(𝑡)𝑑𝑡+_{0≤𝑘≤𝑁}𝑓_𝑘|=|^∞₀𝑓_𝑎(𝑡)𝑑𝑡+^∞_𝑘=0𝑓_𝑘+^∞₀(𝑝_𝑎(𝑡)−𝑓_𝑎(𝑡))𝑑𝑡−_𝑘>𝑁𝑓_𝑘|≤|^∞₀𝑓(𝑡)𝑑𝑡|+‖𝑝_𝑎−𝑓_𝑎‖_𝐿1+_𝑘>𝑁|𝑓_𝑘|=|𝑓(̂‌0)|+‖𝑝_𝑎−𝑓_𝑎‖_𝐿1+_𝑘>𝑁|𝑓_𝑘|<0+𝜖4+𝜖4=𝜖2.(2.22)Thus‖𝑓̂‌−𝑝‖̂_𝒲+=‖𝑓_𝑎−𝑝_𝑎‖_𝐿1+_𝑘>𝑁|𝑓_𝑘|+|𝑓_𝑇|<𝜖4+𝜖4+𝜖2=𝜖.(2.23)This completes the proof.

We are now ready to prove the existence of an approximate identity for the maximal ideal 𝔪₀ in 𝒲⁺.

Proof of Theorem 2.6. We have𝑒_𝑛=𝑠𝑠+1/𝑛=𝑠+1/𝑛−1/𝑛𝑠+1/𝑛=1−1𝑛1𝑠+1/𝑛=1+−1𝑛𝑒^−𝑡/𝑛.(2.24)Thus for an 𝑛∈ℕ,‖𝑒_𝑛‖_𝒲+=‖−1𝑛𝑒^−𝑡/𝑛‖_𝐿1+|1|=1+1=2.(2.25)Given 𝑓̂‌∈𝒲⁺, and 𝜖>0 arbitrarily small, in view of Lemma 2.7, we can find a 𝑝̂∈𝔪₀ such that 𝑝 has compact support and ‖𝑓̂‌−𝑝‖̂_𝒲+<𝜖. Then‖𝑒_𝑛𝑓̂‌−𝑓‖̂‌_𝒲+≤‖𝑒_𝑛𝑝̂−𝑝‖̂_𝒲++‖𝑒_𝑛‖_𝒲+‖𝑓̂‌−𝑝‖̂_𝒲++‖𝑓̂‌−𝑝‖̂_𝒲+.(2.26)So it is enough to prove thatlim_𝑛→∞‖𝑒_𝑛𝑝̂−𝑝‖̂_𝒲+=0(2.27)for all 𝑝̂∈𝔪₀ such that 𝑝 has compact support in [0,∞). We do this below.
We have𝑒_𝑛𝑝̂−𝑝̂=𝑠+1/𝑛−1/𝑛𝑠+1/𝑛𝑝̂−𝑝̂=−1𝑛1𝑠+1/𝑛𝑝̂=−1𝑛(𝑒_−𝑡/𝑛∗𝑝).(2.28)Let 𝐶 denote the convolution 𝑒^−𝑡/𝑛∗𝑝:𝐶(𝑡)∶=^𝑡₀𝑒^{−(𝑡−𝜏)/𝑛}𝑝(𝜏)𝑑𝜏.(2.29)We note that 𝐶∈𝐿¹(0,∞), since 𝐿¹(0,∞) is an ideal in 𝒲⁺. Let 𝑇>0 be such thatsupp(𝑝)⊂[0,𝑇].(2.30)We have‖𝑒_𝑛𝑝̂−𝑝‖̂_𝒲+=1𝑛‖𝐶‖_𝐿1=1𝑛^∞₀|𝐶(𝑡)|𝑑𝑡=1𝑛^𝑇₀|𝐶(𝑡)|𝑑𝑡_{(𝐼)}+1𝑛^∞_𝑇|𝐶(𝑡)|𝑑𝑡_{(𝐼𝐼)}.(2.31)We estimate (𝐼) as follows:(𝐼)=1𝑛^𝑇₀|𝐶(𝑡)|𝑑𝑡=1𝑛^𝑇₀|^𝑡₀𝑒^{−(𝑡−𝜏)/𝑛}𝑝(𝜏)𝑑𝜏|𝑑𝑡≤1𝑛^𝑇₀^𝑡₀𝑒^{−(𝑡−𝜏)/𝑛}|𝑝(𝜏)|𝑑𝜏𝑑𝑡≤1𝑛(^𝑇₀^𝑡₀1⋅|𝑝(𝜏)|𝑑𝜏𝑑𝑡)_{(𝐼𝐼𝐼)}.(2.32)Since the integral (𝐼𝐼𝐼) does not depend on 𝑛, we obtain thatlim_𝑛→∞1𝑛^𝑇₀|𝐶(𝑡)|𝑑𝑡=0.(2.33)Furthermore,(𝐼𝐼)=1𝑛^∞_𝑇|𝐶(𝑡)|𝑑𝑡=1𝑛^∞_𝑇𝑒^−𝑡/𝑛|^𝑡₀𝑒^𝜏/𝑛𝑝(𝜏)𝑑𝜏|𝑑𝑡=1𝑛^∞_𝑇𝑒^−𝑡/𝑛|^∞₀𝑒^𝜏/𝑛𝑝(𝜏)𝑑𝜏|𝑑𝑡(sincesupp(𝑝)⊂[0,𝑇])=1𝑛^∞_𝑇𝑒^−𝑡/𝑛|𝑝(̂−1𝑛)|𝑑𝑡.(2.34)Since 𝑝 has compact support in [0,𝑇], 𝑝̂ is an entire function by the Payley-Wiener theorem (see, e.g., [8, Theorem 7.2.3, page 122]). Consequently,(𝐼𝐼)=1𝑛^∞_𝑇𝑒^−𝑡/𝑛|𝑝(̂−1𝑛)|𝑑𝑡=𝑒^−𝑇/𝑛|𝑝(̂−1𝑛)|→^𝑛→∞1⋅|𝑝(̂0)|=1⋅0=0.(2.35)This completes the proof.

We will also need the following lemma, which is basically a repetition of key steps from Browder's proof of Cohen's factorization theorem; see [9, Theorem 1.6.5, page 74]. We will need this version since in our application in the proof of Theorem 1.3, we are not able to use Cohen's factorization theorem directly.

Lemma 2.8. Let 𝑓₁,𝑓₂∈𝔪₀, and 𝛿>0. Let 𝑈(𝒲⁺) denote the set of all invertible elements in 𝒲⁺. Then there exists a sequence (𝑔_𝑛)_𝑛∈ℕ in 𝒲⁺ such that
(1) for all 𝑛∈ℕ, 𝑔_𝑛∈𝑈(𝒲⁺);(2)(𝑔_𝑛)_𝑛∈ℕ is convergent in 𝒲⁺ to a limit 𝑔∈𝔪₀;(3) for all 𝑛∈ℕ, ‖𝑔⁻¹_𝑛𝑓_𝑖−𝑔⁻¹_𝑛+1𝑓_𝑖‖_𝒲+≤𝛿/2^𝑛, 𝑖=1,2.

Proof. We will first prove two general results in steps (A) and (B), which we will use in the rest of the proof.
(A) Let 𝑒∈𝔪₀ and ‖𝑒‖_𝒲+≤𝐾, where 𝐾>1. Then 1−𝑐+𝑐𝑒∈𝑈(𝒲⁺), where 𝑐 is a number chosen such that0<𝑐<14𝐾<14.(2.36)Indeed,‖𝑐𝑐−1𝑒‖_𝒲+<1/(4𝐾)3/4⋅𝐾=13<1,(2.37)and so(1−𝑐+𝑐𝑒)⁻¹=11−𝑐^∞_𝑘=0(𝑐𝑐−1)^𝑘𝑒^𝑘.(2.38)
(B) Furthermore, under the same assumptions and notation as in (A) above, we now show that if ‖𝑒𝐹−𝐹‖_𝒲+ is small for some 𝐹, then so is ‖𝐸𝐹−𝐹‖_𝒲+, where 𝐸∶=(1−𝑐+𝑐𝑒)⁻¹. Since1=11−𝑐^∞_𝑘=0(𝑐𝑐−1)^𝑘,(2.39)we have‖𝐸𝐹−𝐹‖_𝒲+=‖11−𝑐^∞_𝑘=0(𝑐𝑐−1)^𝑘(𝑒^𝑘𝐹−𝐹)‖_𝒲+≤11−𝑐^∞_𝑘∑₌₀(𝑐1−𝑐)^𝑘‖𝑒^𝑘𝐹−𝐹‖_𝒲+.(2.40)But‖𝑒^𝑘𝐹−𝐹‖_𝒲+=‖∑^𝑘−1_𝑗=0(𝑒^𝑗+1𝐹−𝑒^𝑗𝐹)‖_𝒲+≤^𝑘−1_𝑗=0‖𝑒^𝑗‖_𝒲+‖𝑒𝐹−𝐹‖_𝒲+≤‖𝑒𝐹−𝐹‖^{𝒲+𝑘−1}_𝑗=0‖𝑒‖^𝑗_𝒲+<‖𝑒𝐹−𝐹‖_𝒲+𝐾^𝑘𝐾−1.(2.41)Hence‖𝐸𝐹−𝐹‖_𝒲+<‖𝑒𝐹−𝐹‖_𝒲+11−𝑐^∞_𝑘=01𝐾−1(14(1−𝑐))^𝑘<2𝐾−1‖𝑒𝐹−𝐹‖_𝒲+.(2.42)This estimate will be used in constructing the sequence of 𝑔_𝑛's.
Let (𝑒_𝑛)_𝑛∈ℕ denote the approximate identity for 𝔪₀ from Theorem 2.6. Let 𝐾>1 be such that ‖𝑒_𝑛‖_𝒲+≤𝐾 for all 𝑛∈ℕ. Choose 𝑐 such that0<𝑐<14𝐾<14.(2.43)
We will inductively define a sequence (𝑒_𝑚𝑘)_𝑘∈ℕ with terms from the approximate identity for 𝔪₀ such that if𝑔_𝑛∶=𝑐^𝑛_𝑘=1(1−𝑐)^𝑘−1𝑒_𝑚𝑘+(1−𝑐)^𝑛,(2.44)then we have ‖𝑓_𝑖−𝑔⁻¹₁𝑓_𝑖‖_𝒲+<𝛿/2, 𝑖=1,2, and
(P1) for all 𝑛∈ℕ, 𝑔_𝑛∈𝑈(𝒲⁺),(P2)for all 𝑛∈ℕ, ‖𝑔⁻¹_𝑛𝑓_𝑖−𝑔⁻¹_𝑛+1𝑓_𝑖‖_𝒲+<𝛿/2^𝑛, 𝑖=1,2. Since (𝑒_𝑛)_𝑛∈ℕ is an approximate identity for 𝔪₀, we can choose 𝑚₁ such that‖𝑒_𝑚1𝑓_𝑖−𝑓_𝑖‖_𝒲+<𝛿4(𝐾−1),𝑖=1,2.(2.45)Define 𝑔₁=𝑐𝑒_𝑚1+1−𝑐. So by (A), 𝑔₁∈𝑈(𝒲⁺) and using the calculation in (B), we see that‖𝑓_𝑖−𝑔⁻¹₁𝑓_𝑖‖_𝒲+<𝛿2,𝑖=1,2.(2.46)Suppose that 𝑒_𝑚1,…,𝑒_𝑚𝑛 have been constructed, so that 𝑔_𝑛 defined by (2.44) satisfies (P1) and (P2). We assert that if we choose 𝑒_𝑚𝑛+1 such that‖𝑒_𝑚𝑛+1𝑓_𝑖−𝑓_𝑖‖_𝒲+(𝑖=1,2),‖𝑒_𝑚𝑛+1𝑒_𝑚𝑘−𝑒_𝑚𝑘‖_𝒲+(1≤𝑘≤𝑛)(2.47)are sufficiently small, then 𝑔_𝑛+1 defined by (2.44) satisfies (P1) and (P2), completing the induction step.
Indeed, if 𝐸∶=(1−𝑐+𝑐𝑒_𝑚𝑛+1)⁻¹, we have𝑔_𝑛=𝐸⁻¹𝑐^𝑛_𝑘=1(1−𝑐)^𝑘−1𝐸𝑒_𝑚𝑘+(1−𝑐)^𝑛,𝑔_𝑛+1=𝐸⁻¹(𝑐^𝑛_𝑘=1(1−𝑐)^𝑘−1𝐸𝑒_𝑚𝑘+(1−𝑐)^𝑛).(2.48)Let 𝐺_𝑛 be defined by𝐺_𝑛=𝑐^𝑛_𝑘=1(1−𝑐)^𝑘−1𝐸𝑒_𝑚𝑘+(1−𝑐)^𝑛.(2.49)Then we have‖𝐺_𝑛−𝑔_𝑛‖_𝒲+<𝑐∑^𝑛_𝑘=1(1−𝑐)^𝑘−1‖𝐸𝑒_𝑚𝑘−𝑒_𝑚𝑘‖_𝒲+<max_{1≤𝑘≤𝑛}‖𝐸𝑒_𝑚𝑘−𝑒_𝑚𝑘‖_𝒲+<2𝐾−1max_{1≤𝑘≤𝑛}‖𝑒_𝑚𝑛+1𝑒_𝑚𝑘−𝑒_𝑚𝑘‖_𝒲+.(2.50)Hence 𝐺_𝑛∈𝑈(𝒲⁺) and moreover ‖𝐺⁻¹_𝑛−𝑔⁻¹_𝑛‖_𝒲+ is small, provided only that ‖𝑒_𝑚𝑛+1𝑒_𝑚𝑘−𝑒_𝑚𝑘‖_𝒲+ is small for 𝑘=1,…,𝑛. (Indeed, this is because 𝑈(𝒲⁺) is an open set in 𝒲⁺.)
Since 𝑔_𝑛+1=𝐸⁻¹𝐺_𝑛, we have then 𝑔_𝑛+1∈𝑈(𝒲⁺), 𝑔⁻¹_𝑛+1=𝐺⁻¹_𝑛𝐸, and so for 𝑖=1,2,‖𝑔⁻¹_𝑛+1𝑓_𝑖−𝑔⁻¹_𝑛𝑓_𝑖‖_𝒲+=‖𝐺⁻¹_𝑛𝐸𝑓_𝑖−𝑔⁻¹_𝑛𝑓_𝑖‖_𝒲+≤‖𝐺⁻¹_𝑛𝐸𝑓_𝑖−𝑔⁻¹_𝑛𝐸𝑓_𝑖‖_𝒲++‖𝑔⁻¹_𝑛𝐸𝑓_𝑖−𝑔⁻¹_𝑛𝑓_𝑖‖_𝒲+≤‖𝐺⁻¹_𝑛−𝑔⁻¹_𝑛‖_𝒲+‖𝐸𝑓_𝑖‖_𝒲++‖𝑔⁻¹_𝑛‖_𝒲+‖𝐸𝑓_𝑖−𝑓_𝑖‖_𝒲+.(2.51)Moreover, recall that by (B), we know that‖𝐸𝑓_𝑖−𝑓_𝑖‖_𝒲+≤2𝐾−1‖𝑒_𝑚𝑛+1𝑓_𝑖−𝑓_𝑖‖_𝒲+,𝑖=1,2.(2.52)Thus if ‖𝑒_𝑚𝑛+1𝑓_𝑖−𝑓_𝑖‖_𝒲+ (𝑖=1,2) and ‖𝑒_𝑚𝑛+1𝑒_𝑚𝑘−𝑒_𝑚𝑘‖_𝒲+ (1≤𝑘≤𝑛) are sufficiently small, we will have ‖𝑔⁻¹_𝑛+1𝑓_𝑖−𝑔⁻¹_𝑛𝑓_𝑖‖_𝒲+ as small as we please. This completes the induction step.
Since ‖𝑒_𝑚𝑘‖_𝒲+≤𝐾, 0<1−𝑐<1, and 𝒲⁺ is a Banach algebra, it follows that𝑔_𝑛→𝑐^∞_𝑘=1(1−𝑐)^𝑘−1𝑒_𝑚𝑘=∶𝑔∈𝔪₀,(2.53)and the proof is completed.

3. Noncoherence of 𝒲⁺

Proof of Theorem 1.3. We will use the characterization that an integral domain is coherent if and only if the intersection of any two finitely generated ideals of the ring is again finitely generated; see [1, Theorem 2.3.2, page 45]. In fact, we present two finitely generated ideals 𝐼 and 𝐽 such that 𝐼∩𝐽 is not finitely generated.
Let 𝑝,𝑆 be given by𝑝=(1−𝑒^−𝑠)³,𝑆=𝑒^{−(1+𝑒−𝑠)/(1−𝑒−𝑠)}.(3.1)Clearly we have 𝑝∈𝔪₀.
It is known (see, e.g., [3, Remark after Theorem 1, page 224]) that(1−𝑧)³𝑒^{−(1+𝑧)/(1−𝑧)}∈𝑊⁺(𝔻)∶={𝑓(𝑧)=^∞_𝑛=0𝑎_𝑛𝑧^𝑛(𝑧∈𝔻)|^∞_𝑛=0|𝑎_𝑛|<∞}.(3.2)Here 𝔻∶={𝑧∈ℂ∣|𝑧|≤1}. So if 𝑎_𝑛's are defined via(1−𝑧)³𝑒^{−(1+𝑧)/(1−𝑧)}=𝑎₀+𝑎₁𝑧+𝑎₂𝑧²+𝑎₃𝑧³+⋯,𝑧∈𝔻,(3.3)then we have^∞_𝑘=0|𝑎_𝑘|<∞.(3.4)If Re(𝑠)>0, then 𝑒^−𝑠∈𝔻, and so from (3.3), we have𝑝𝑆=𝑎₀+𝑎₁𝑒^−𝑠+𝑎₂𝑒^−2𝑠+𝑎₃𝑒^−3𝑠+⋯,Re(𝑠)>0.(3.5)Since ∑^∞_𝑘=0|𝑎_𝑘|<∞, the right-hand side in (3.5) belongs to 𝒲⁺. So 𝑝𝑆∈𝒲⁺.
We define the ideals 𝐼=(𝑝) and 𝐽=(𝑝𝑆) of 𝒲⁺.
Let𝐾∶={𝑝𝑆𝑓∣𝑓∈𝒲⁺and𝑆𝑓∈𝒲⁺}.(3.6)We claim that 𝐾=𝐼∩𝐽. Trivially 𝐾⊂𝐼∩𝐽. To prove the reverse inclusion, let 𝑔∈𝐼∩𝐽. Then there exist two functions 𝑓 and ℎ in 𝒲⁺ such that 𝑔=𝑝ℎ=𝑝𝑆𝑓. Since 𝑝≠0 and 𝒲⁺ is an integral domain, we obtain that 𝑆𝑓=ℎ∈𝒲⁺. So 𝑔∈𝐾.
Let 𝐿 denote the ideal𝐿∶={𝑓∈𝒲⁺∣𝑆𝑓∈𝒲⁺}.(3.7)Then 𝐾∶=𝑝𝑆𝐿. Since 𝑆 has a singularity at 𝑠=0, it follows that 𝐿⊂𝔪₀. We will show that 𝐿=𝐿𝔪₀. Let 𝑓∈𝐿. We would like to factor 𝑓=ℎ𝑔 with ℎ∈𝐿 and 𝑔∈𝔪₀. Applying Lemma 2.8 with 𝑓₁∶=𝑓∈𝔪₀ and 𝑓₂∶=𝑆𝑓∈𝔪₀, for any 𝛿>0, there exists a sequence (𝑔_𝑛)_𝑛∈ℕ in 𝒲⁺ such that
(1) for all 𝑛∈ℕ, 𝑔_𝑛∈𝑈(𝒲⁺);(2)(𝑔_𝑛)_𝑛∈ℕ is convergent in 𝒲⁺ to a limit 𝑔∈𝔪₀;(3) for all 𝑛∈ℕ,‖𝑔⁻¹_𝑛𝑓−𝑔⁻¹_𝑛+1𝑓‖_𝒲+≤𝛿2^𝑛,‖𝑔⁻¹_𝑛𝑆𝑓−𝑔⁻¹_𝑛+1𝑆𝑓‖_𝒲+≤𝛿2^𝑛.(3.8) Putℎ_𝑛∶=𝑔⁻¹_𝑛𝑓,𝐻_𝑛∶=𝑔⁻¹_𝑛𝑆𝑓.(3.9)Then ℎ_𝑛∈𝔪₀. Also 𝐻_𝑛∈𝔪₀, since |𝑆| is bounded by 1 on Re(𝑠)>0 and 𝑓(0)=0. The estimates above imply that (ℎ_𝑛)_𝑛∈ℕ and (𝐻_𝑛)_𝑛∈ℕ are Cauchy sequences in 𝒲⁺. Since 𝔪₀ is closed, they converge to elements ℎ and 𝐻, respectively, in 𝔪₀, that is, ℎ_𝑛=𝑔⁻¹_𝑛𝑓→ℎ and 𝐻_𝑛=𝑔⁻¹_𝑛𝑆𝑓=𝑆ℎ_𝑛→𝐻. Since convergence in 𝒲⁺ implies convergence in 𝐻^∞ (Lemma 2.2), it follows thatℎ_𝑛→𝐻^∞ℎ(sinceℎ_𝑛→𝒲⁺ℎ),𝑆ℎ_𝑛→𝐻^∞𝑆ℎ(sinceℎ_𝑛→𝐻^∞ℎ,𝑆∈𝐻^∞),𝑆ℎ_𝑛→𝐻^∞𝐻(since𝐻_𝑛→𝒲⁺𝐻)(3.10)and so by the uniqueness of the limit of the sequence (𝑆ℎ_𝑛)_𝑛∈ℕ in 𝐻^∞, we have 𝑆ℎ=𝐻. Also, in 𝒲⁺-norm we have𝑓=lim_𝑛→∞ℎ_𝑛𝑔_𝑛=ℎ𝑔,(3.11)since multiplication is continuous in the Banach algebra 𝒲⁺. Since ℎ and 𝑆ℎ=𝐻 belong to 𝔪₀⊂𝒲⁺, we see that ℎ∈𝐿. Moreover, as 𝑔∈𝔪₀, we have got the desired factorization and 𝐿=𝐿𝔪₀.
But 𝐿≠(0), since 𝑝∈𝐿. By Lemma 2.3, it follows that 𝐿 cannot be finitely generated. Therefore, 𝑝𝑆𝐿=𝐼∩𝐽 is not finitely generated.

Remark 3.1. The ideal 𝐿 in the above proof can be interpreted as an ideal of denominators; see [10, page 396]. Using the fact that 𝑝𝑆∈𝒲⁺, we have 𝑆∈𝑄(𝒲⁺), where 𝑄(𝒲⁺) denotes the field of fractions of 𝒲⁺. We can then consider the fractional ideal 𝑀∶=𝒲⁺+𝒲⁺𝑆 of 𝒲⁺ (see [11, page 19]) and the ideal of denominators 𝐿 of 𝑆, namely 𝐿=𝒲⁺∶𝑀={𝑑∈𝒲⁺∣𝑑𝑆∈𝒲⁺}.
Based on the results in [12, Theorem 3, Example 3], it follows that 𝑆∈𝑄(𝒲⁺) does not admit a weak coprime factorization, since 𝐿 is not a principal ideal of 𝒲⁺. In particular, 𝑆 does not admit a coprime factorization, that is, there do not exist 𝑑,𝑥,𝑦,𝑛∈𝒲⁺ such that 𝑑≠0, 𝑆=𝑛/𝑑, and 𝑑𝑥−𝑛𝑦=1. Moreover, 𝑆 is not internally stabilizable, since otherwise 𝐿 would be generated by two elements. Finally, the fact that 𝐿 is not finitely generated implies that 𝒲⁺ is not a greatest common divisor domain: indeed, were it the case that 𝒲⁺ is a greatest common divisor domain, then by [12, Corollary 3], every element in 𝑄(𝒲⁺) would admit a weak coprime factorization.

Acknowledgment

The author thanks all the referees for their careful review, and in particular, two of the referees for the Remarks 2.4 and 3.1.