Abstract

Let 𝐶(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(𝑥,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={𝑧∈ℂ∶|𝑧|<1} is the unit disc in the complex plane ℂ, and have 𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐶𝐴(𝑛)=𝐶𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐶(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐶𝐴(𝑛)=𝐶(𝑛)(𝔻×𝔻)∩ℋ𝑜𝑙(𝔻×𝔻)). The double integration operator is defined in 𝐶𝐴(𝑛) by the formula ∫𝑊𝑓(𝑧,𝑤)=𝑧0∫𝑤0𝑓(𝑢,𝑣)𝑑𝑣𝑑𝑢. By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator 𝑊∣𝐸𝑧𝑤, where 𝐸𝑧𝑤={𝑓∈𝐶𝐴(𝑛)∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤)} is an invariant subspace of 𝑊, and study its properties. We also study invertibility of the elements in 𝐶𝐴(𝑛) with respect to the Duhamel product.

1. Introduction and Backgrounds

Let ℋ𝑜𝑙(𝔻×𝔻) denote the Fréchet space of functions 𝑓(𝑧,𝑤) that are holomorphic in the bidisc 𝔻×𝔻={(𝑧,𝑤)∈ℂ×ℂ∶|𝑧|<1and|𝑤|<1}. The product we define on this space is 𝜕(𝑓⊛𝑔)(𝑧,𝑤)∶=2𝜕𝑧𝜕𝑤𝑧0𝑤0𝑓(𝑧−𝑢,𝑤−𝑣)𝑔(𝑢,𝑣)𝑑𝑣𝑑𝑢,(1.1) which obviously defines an integrodifferential operator 𝒟𝑓,𝒟𝑓𝑔∶=𝑓⊛𝑔. This product is a natural extension of the Duhamel product on ℋ𝑜𝑙(𝔻) [1]: 𝑑(𝑓⊛𝑔)(𝑧)∶=𝑑𝑧z0𝑓(𝑧−𝑡)𝑔(𝑡)𝑑𝑡=z0ğ‘“î…ž(𝑧−𝑡)𝑔(𝑡)𝑑𝑡+𝑓(0)𝑔(𝑧),(1.2) where the integrals are taken over the segment joining the points 0 and 𝑧.

Note that the Duhamel product is widely applied in various questions of analysis, for example, in the theory of differential equations and in solution of boundary value problems of mathematical physics. Wigley [2] showed that, for 𝑝≥1, the Hardy space 𝐻𝑝(𝔻) (which is the space of all holomorphic functions on the open unit disc 𝔻 for which the norm ‖𝑓‖𝐻𝑝=sup0<𝑟<112𝜋02𝜋||𝑓𝑟𝑒𝑖𝜃||𝑝𝑑𝜃1/𝑝(1.3) is finite) is a Banach algebra under the Duhamel product ⊛.

The Hardy space of the polydisc, 𝐻𝑝(𝔻𝑛), is defined as those functions analytic on 𝔻𝑛∶=𝔻×⋯×𝔻 for which the following norm is finite: ‖𝑓‖𝐻𝑝∶=sup𝑟1<1⋯sup𝑟𝑛<11(2𝜋)𝑛02𝜋⋯02𝜋||𝑓𝑟1𝑒𝑖𝜃1,…,𝑟𝑛𝑒𝑖𝜃𝑛||𝑝𝑑𝜃1⋯𝑑𝜃𝑛1/𝑝.(1.4) If 𝑝≥1, this is a Banach space, and if 0<𝑟<1, this is a Fréchet space [3]. In [3], Merryfield and Watson proved that for 𝑝≥1𝐻𝑝(𝔻𝑛) is a Banach algebra with respect to the product (1.1).

In the present paper we prove that the space 𝐸𝑧𝑤 can be given a Banach algebra structure under the Duhamel product (1.1); in particular, we describe the maximal ideal space of the Banach algebra (𝐸𝑧𝑤,⊛), where 𝐸𝑧𝑤∶=𝑓∈𝐶𝐴(𝑛)𝜕∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤),(1.5)(𝑓⊛𝑔)(𝑧𝑤)=2𝜕𝑧𝜕𝑤𝑧0𝑤0𝑓((𝑧−𝑢)(𝑤−𝑣))𝑔(𝑢𝑣)𝑑𝑣𝑑𝑢.(1.6) By using product (1.6) we also describe commutant of the operator 𝑊𝑧𝑤∶=𝑊∣𝐸𝑧𝑤, that is, the set of bounded linear operators on 𝐸𝑧𝑤 commuting with 𝑊𝑧𝑤. Moreover, we describe the set of cyclic vectors of the double integration operator 𝑊𝑧𝑤 acting on the closed subspace 𝐻𝑝𝑧𝑤∶=𝑓(𝑧,𝑤)∈𝐻𝑝𝔻2.∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤)(1.7) We recall that a vector 𝑥∈𝑋 is called cyclic vector for the operator 𝐴∈ℒ(𝑋) (Banach algebra of all bounded linear operators on a Banach space 𝑋) if span𝑥,𝐴𝑥,𝐴2𝑥,…=𝑋,(1.8) where span{𝑥,𝐴𝑥,𝐴2𝑥,…} denotes the closure of the linear hull of the set {𝑥,𝐴𝑥,𝐴2𝑥,…}.

2. Description of {𝑊𝑧𝑤}

For any operator 𝐴∈ℒ(𝑋) its commutant {𝐴} is defined by {𝐴}∶={𝐵∈ℒ(𝑋)∶𝐵𝐴=𝐴𝐵}.(2.1)

The study of commutant of the concrete operator 𝐴∈ℒ(𝑋) is one of the important, but generally, not easy problem of operator theory. For this, it is enough to remember the famous Lomonosov’s theorem on the existence of nontrivial hyperinvariant subspace of compact operator 𝒦 on a Banach space 𝑋 (recall that a closed subspace 𝐸⊂𝑋 is called hyperinvariant subspace for the operator 𝐴∈ℒ(𝑋), if it is invariant for any operator 𝐵∈{𝐴}). Note that many papers are devoted to the evident description of commutant (and, more generally, the set of so-called extended eigenvectors [4–6]) for some special operator classes (see, e.g., [7–14]). In this section we describe in terms of the Duhamel operators the commutant of the operator 𝑊𝑧𝑤 on the closed subspace 𝐸𝑧𝑤 of the space 𝐶𝐴(𝑛). First, we prove the following lemma, which shows that 𝐸𝑧𝑤 is a Banach algebra under the Duhamel product ⊛ given by formula (1.6).

Lemma 2.1. (𝐸𝑧𝑤,⊛) is a Banach algebra.

Proof. Indeed, let 𝑓,𝑔∈𝐸𝑧𝑤 be two functions. The norm in 𝐸𝑧𝑤 is defined by ‖𝑓‖𝑛∶=maxmax(𝑧,𝑤)∈𝔻2||||𝜕|𝛼|𝑓(𝑧𝑤)𝜕𝑧𝛼1𝜕𝑤𝛼2||||∶|𝛼|=𝛼1+𝛼2.=0,1,…,𝑛(2.2) Using (1.6), (2.2) and the Leibnitz formula for the partial derivatives of the product 𝑓⊛𝑔, it can be proved (which is omitted) that (see e.g., the method of the paper [15, 16]) ‖𝑓⊛𝑔‖𝑛≤𝐶𝑛‖𝑓‖𝑛‖𝑔‖𝑛(2.3) for some constant 𝐶𝑛>0, which proves the lemma.

The main result of this section is the following theorem.

Theorem 2.2. Let 𝑇∈ℒ(𝐸𝑧𝑤) be an operator. Then 𝑇∈{𝑊𝑧𝑤} if and only if there exists a function 𝜑∈𝐸𝑧𝑤 such that 𝑇=𝒟𝜑, where 𝒟𝜑 is the Duhamel operator defined by 𝒟𝜑𝜕𝑓(𝑧𝑤)=(𝜑⊛𝑓)(𝑧𝑤)=2𝜕𝑧𝜕𝑤𝑧0𝑤0𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢.(2.4)

Proof. Let 𝑇∈{𝑊𝑧𝑤}, that is, 𝑇𝑊𝑧𝑤=𝑊𝑧𝑤𝑇(2.5) Then we have that 𝑇𝑊𝑧𝑤(𝑧𝑤)𝑘=𝑊𝑧𝑤𝑇(𝑧𝑤)𝑘,(2.6) for all 𝑘=0,1,…, whence by computing 𝑊𝑧𝑤(𝑧𝑤)𝑘 we have 𝑇𝑧0𝑤0(𝑢𝑣)𝑘𝑑𝑣𝑑𝑢=𝑇𝑧0𝑢𝑘𝑤0𝑣𝑘𝑑𝑣𝑑𝑢=𝑇𝑧0𝑢𝑘𝑤𝑘+1𝑧𝑘+1𝑑𝑢=𝑇𝑘+1𝑤𝑘+1(𝑘+1)2=1(𝑘+1)2𝑇(𝑧𝑤)𝑘+1,(2.7) or 𝑇(𝑧𝑤)𝑘+1=(𝑘+1)2𝑊𝑧𝑤𝑇(𝑧𝑤)𝑘,(2.8) for all 𝑘=0,1,….
From (2.8) we have by induction that 𝑇(𝑧𝑤)𝑘=𝑊𝑘𝑧𝑤𝑇1𝑘𝑠=1𝑠2(𝑘=1,2,…).(2.9) Indeed, for 𝑘=1 we have from (2.8) 𝑇(𝑧𝑤)=𝑊𝑧𝑤𝑇1, as desired.
Assume for 𝑘=𝑛 that 𝑇(𝑧𝑤)𝑛=𝑊𝑛𝑧𝑤𝑇1𝑛𝑠=1𝑠2.(2.10) For 𝑘=𝑛+1 we have from (2.8) that 𝑇(𝑧𝑤)𝑛+1=(𝑛+1)2𝑊𝑧𝑤𝑇(𝑧𝑤)𝑛.(2.11) Now, by considering (2.10) from the latter equality we have 𝑇(𝑧𝑤)𝑛+1=(𝑛+1)2𝑊𝑧𝑤𝑊𝑛𝑧𝑤𝑇1𝑛𝑠=1𝑠2=𝑊𝑛+1𝑧𝑤𝑇1(𝑛+1)2𝑛𝑠=1𝑠2=𝑊𝑛+1𝑧𝑤𝑇1𝑛+1𝑠=1𝑠2,(2.12) which proves (2.9).
Now, let us show that 𝑊𝑘𝑧𝑤𝑓(𝑧𝑤)=𝑧0𝑤0[](𝑧−𝑢)(𝑤−𝑣)𝑘−1[](𝑘−1)!2𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢.(2.13) For this purpose, first show that 𝑊𝑘𝑧𝑤𝑓(𝑧𝑤)=(𝑧𝑤)𝑘[]𝑘!2⊛𝑓(𝑧𝑤),(2.14) for all 𝑘≥0. Indeed, it follows directly from (1.6) that 1 is the unit with respect to the Duhamel product ⊛ in 𝐸𝑧𝑤, and 𝑊𝑧𝑤𝑓=𝑧𝑤⊛𝑓(𝑧𝑤) for every 𝑓∈𝐸𝑧𝑤. From this by induction we have equality (2.14) (we omit details).
Then we have 𝑊𝑘𝑧𝑤𝑓(𝑧𝑤)=(𝑧𝑤)𝑘[]𝑘!2𝜕⊛𝑓(𝑧𝑤)=2𝜕𝑧𝜕𝑤𝑧0𝑤0[](𝑧−𝑢)(𝑤−𝑣)𝑘[](𝑘!)2=1𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢(𝑘!)2𝑧0𝑤0𝑘2[](𝑧−𝑢)(𝑤−𝑣)𝑘−1𝑓=𝑘(𝑢𝑣)𝑑𝑣𝑑𝑢2𝑘2[](𝑘−1)!2𝑧0𝑤0[](𝑧−𝑢)(𝑤−𝑣)𝑘−1=𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑧0𝑤0[](𝑧−𝑢)(𝑤−𝑣)𝑘−1[](𝑘−1)!2𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢,(2.15) which proves (2.13).
Now by combining (2.9) and (2.13) we have 𝑇(𝑧𝑤)𝑘=𝑘𝑠=1𝑠2𝑧0𝑤0[](𝑧−𝑢)(𝑤−𝑣)𝑘−1[](𝑘−1)!2𝑇1𝑑𝑣𝑑𝑢,(2.16) for all 𝑘≥0, which means that 𝑇(𝑧𝑤)𝑘=(𝑧𝑤)𝑘⊛𝑇1(𝑘≥0),(2.17) and hence 𝑇𝑝(𝑧𝑤)=𝑝(𝑧𝑤)⊛𝑇1,(2.18) for all polynomials 𝑝. Thus, by Lemma 2.1 and Weierstrass approximation theorem, we deduce that 𝜕(𝑇𝑓)(𝑧𝑤)=𝑇1⊛𝑓(𝑧𝑤)=2𝜕𝑧𝜕𝑤𝑧0𝑤0=𝜕𝑓((𝑧−𝑢)(𝑤−𝑣))(𝑇1)(𝑢𝑣)𝑑𝑣𝑑𝑢2𝜕𝑧𝜕𝑤𝑧0𝑤0=(𝑇1)((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑧0𝑤0𝜕2=𝜕𝑧𝜕𝑤(𝑇1)((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑧0𝑤0(𝑧−𝑢)(𝑤−𝑣)(𝑇1)𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑇1)𝑤=((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢+(𝑇1)(0)𝑓(𝑧𝑤)𝑧0𝑤0𝜑𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢+𝜑0𝑓(𝑧𝑤),(2.19) where 𝜑∶=𝑇1∈𝐸𝑧𝑤.(2.20) Thus, (𝑇𝑓)(𝑧𝑤)=𝜑(𝑧𝑤)⊛𝑓(𝑧𝑤)=𝒟𝜑𝑓(𝑧𝑤)(2.21) for all 𝑓∈𝐸𝑧𝑤 and some 𝜑∈𝐸𝑧𝑤. Conversely, if 𝜑∈𝐸𝑧𝑤, then 𝒟𝜑 commutes with 𝑊𝑧𝑤. Since by Lemma 2.1𝐸𝑧𝑤 is a Banach algebra with respect to the Duhamel product ⊛, 𝒟𝜑 is bounded operator on 𝐸𝑧𝑤. The theorem is proved.

Corollary 2.3. For a function 𝜑 in 𝐸𝑧𝑤 there exists a unique commutant 𝑇 of the operator 𝑊𝑧𝑤 such that 𝑇1=𝜑(𝑧𝑤).

Corollary 2.4. One has {𝑊𝑧𝑤}={𝑊𝑧𝑤}, where {𝑊𝑧𝑤} stands for the bicommutant of the operator 𝑊𝑧𝑤.

Proof. It suffices to prove that 𝑇1𝑇2=𝑇2𝑇1 for every 𝑇1, 𝑇2 in {𝑊𝑧𝑤}. Indeed, by Theorem 2.2, there exist 𝜑,𝜓∈𝐸𝑧𝑤 such that 𝑇1𝑓(𝑧𝑤)=𝜑(0)𝑓(𝑧𝑤)+𝑧0𝑤0𝜑𝑤+((𝑧−𝑢)(𝑤−𝑣))(𝑧−𝑢)(𝑤−𝑣)𝜑𝑧𝑤𝑓=((𝑧−𝑢)(𝑤−𝑣))(𝑢𝑣)𝑑𝑣𝑑𝑢𝜑(0)𝐼+𝒦Φ𝑇𝑓(𝑧𝑤),(2.22)2𝑓(𝑧𝑤)=𝜓(0)𝑓(𝑧𝑤)+𝑧0𝑤0𝜓𝑤+((𝑧−𝑢)(𝑤−𝑣))(𝑧−𝑢)(𝑤−𝑣)𝜓𝑧𝑤𝑓=((𝑧−𝑢)(𝑤−𝑣))(𝑢𝑣)𝑑𝑣𝑑𝑢𝜓(0)𝐼+𝒦Ψ𝑓(𝑧𝑤)(2.23) for all 𝑓∈𝐸𝑧𝑤, where 𝜕Φ(𝑧𝑤)∶=2=𝜕𝜕𝑧𝜕𝑤𝜑(𝑧𝑤),Ψ(𝑧𝑤)∶2𝒦𝜕𝑧𝜕𝑤𝜓(𝑧𝑤),Φ𝑓(𝑧𝑤)=(Φ∗𝑓)(𝑧𝑤)∶=𝑧0𝑤0𝜕2=𝜕𝑧𝜕𝑤𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑧0𝑤0𝜑𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑𝑧𝑤𝒦((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢,Ψ𝑓(𝑧𝑤)=(Ψ∗𝑓)(𝑧𝑤)∶=𝑧0𝑤0𝜕2𝜓=𝜕𝑧𝜕𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑧0𝑤0𝜓𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜓𝑧𝑤𝑓((𝑧−𝑢)(𝑤−𝑣))(𝑢𝑣)𝑑𝑣𝑑𝑢.(2.24) Since the usual convolution operators 𝒦Φ and 𝒦Ψ are commuting operators, we have 𝑇1𝑇2=𝜑(0)𝐼+𝒦Φ𝜓(0)𝐼+𝒦Ψ=𝜓(0)𝐼+𝒦Ψ𝜑(0)𝐼+𝒦Φ=𝑇2𝑇1,(2.25) which proves the corollary.

Theorem 2.5. An operator 𝑇∈ℒ(𝐸𝑧𝑤) will be an isomorphism of the space 𝐸𝑧𝑤 into itself and commutes with 𝑊𝑧𝑤 if and only if it can be written in the form (𝑇𝑓)(𝑧𝑤)=𝜑(0)𝑓(𝑧𝑤)+𝑧0𝑤0𝜑𝑤+((𝑧−𝑢)(𝑤−𝑣))(𝑧−𝑢)(𝑤−𝑣)𝜑𝑧𝑤𝑓((𝑧−𝑢)(𝑤−𝑣))(𝑢𝑣)𝑑𝑣𝑑𝑢,(2.26) and𝜑(0)=(𝑇1)∣𝑧𝑤=0≠0.

Proof. If 𝑇∈ℒ(𝐸𝑧𝑤) is an isomorphism of the space 𝐸𝑧𝑤 into itself and commutes with 𝑊𝑧𝑤, then by Theorem 2.2 we have for 𝑇 representation (2.26) with 𝑇𝑊𝑧𝑤=𝑊𝑧𝑤𝑇. Clearly, it follows from this equality and (2.26) that 𝜑(0)=(𝑇1)∣𝑧𝑤=0≠0.
Conversely, suppose that 𝑇 has the form (2.26) with 𝜑(0)=(𝑇1)∣𝑧𝑤=0≠0, and prove then that 𝑇∈{𝑊𝑧𝑤} and 𝑇 is an isomorphism on 𝐸𝑧𝑤. Indeed, the inclusion 𝑇∈{𝑊𝑧𝑤} follows directly from Theorem 2.2. On the other hand, it is easy to see from the representation (2.26) that 𝑇=𝒟𝜑=𝜑(0)𝐼+𝒦Φ,(2.27) where 𝒦Φ, 𝒦Φ𝑓=𝑧0𝑤0𝜑𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢,(2.28) is the usual convolution operator on 𝐸𝑧𝑤. It is not difficult to see that 𝒦Φ is a compact operator on 𝐸𝑧𝑤.
Let us show that ker𝒟𝜑={0}. Indeed, let 𝒟𝜑𝑓=0, where 𝑓∈𝐸𝑧𝑤. Then, (𝜑(0)𝐼+𝒦Φ)𝑓=0, that is, 𝜕2𝜕𝑧𝜕𝑤𝑧0𝑤0𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢=0.(2.29) By standard calculation, we obtain from (2.29) that 𝑧0𝑤0𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢=𝑐1𝑧+𝑐2,(2.30) where 𝑐1, 𝑐2 are constants. Since 0=𝑧0𝑤0𝜑||||((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑧=0=𝑐1𝑧+𝑐2||𝑧=0=𝑐2,(2.31) we have that 𝑐2=0. On the other hand, since 0=𝑧0𝑤0𝜑||||((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑤=0=𝑐1𝑧,(2.32) for all 𝑧∈𝔻, we obtain that 𝑐1=0. Thus, 𝑧0𝑤0𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢=0,(2.33) for all 𝑧∈𝔻 and 𝑤∈𝔻. Now, by considering that 𝜑(0)≠0 and 𝜑 is a continuous function on 𝔻×𝔻, by the Titchmarsh Convolution Theorem [17] for functions of several variables we deduce from (2.33) that 𝑓(𝑧𝑤)=0 for all 𝑧,𝑤∈𝔻, that is, ker𝑇={0}. Since 𝒦Φ is compact, it follows from Fredholm alternative that 𝑇 is invertible in 𝐸𝑧𝑤, that is, 𝑇 is an isomorphism. The theorem is proved.

From Theorem 2.5 and Corollary 2.3 we obtain the following.

Corollary 2.6. For any function 𝜑 belonging to 𝐸𝑧𝑤 and satisfying 𝜑(0)≠0, there exists a unique isomorphism 𝑇 of the space 𝐸𝑧𝑤 such that 𝑇 commutes with 𝑊𝑧𝑤 and 𝑇1=𝜑(𝑧𝑤).

Corollary 2.7. If 𝜑∈𝐸𝑧𝑤 and 𝜑(0)≠0, then the integrodifferential equation 𝜑(0)𝑥(𝑧𝑤)+𝑧0𝑤0𝜑𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑥(𝑢𝑣)𝑑𝑣𝑑𝑢=𝑦(𝑧𝑤)(2.34) has a unique solution for any right-hand side 𝑦∈𝐸𝑧𝑤.

Corollary 2.8. The unique maximal ideal of the Banach algebra (𝐸𝑧𝑤,⊛) is {𝑓∈𝐸𝑧𝑤∶𝑓(0)=0}; that is, the maximal ideal space of (𝐸𝑧𝑤,⊛) consists of one homomorphizm, namely evaluation at the origin ℎ(𝑓)=𝑓(0).

3. Cyclic Vectors of 𝑊𝑧𝑤

Let us consider the restricted operator 𝑊𝑧𝑤=𝑊∣𝐻𝑝𝑧𝑤. In this section we will describe the set of all cyclic vectors of this operator. The main result of this section is the following.

Theorem 3.1. Let 𝑓∈𝐻𝑝𝑧𝑤. Then 𝑊span𝑛𝑧𝑤𝑓∶𝑛=0,1,2,…=𝐻𝑝𝑧𝑤,(3.1) if and only if 𝑓∣𝑧𝑤=0≠0.

Proof. It is easy to verify that 𝑊𝑘𝑧𝑤𝑔(𝑧𝑤)=(𝑧𝑤)𝑘(𝑘!)2⊛𝑔(𝑧𝑤)(𝑘≥0),(3.2) for all 𝑔∈𝐻𝑝𝑧𝑤. Let us define the integrodifferential operator (or, briefly, the Duhamel operator) 𝒟𝑓 defined by (𝒟𝑓𝑔)(𝑧𝑤)=(𝑓⊛𝑔)(𝑧𝑤),𝑔∈𝐻𝑝𝑧𝑤. By the known result of Merryfield and Watson (see [3, Corollary 2.6]), 𝐻𝑝(𝔻2),𝑝≥1, is the Banach algebra with respect to the Duhamel product ⊛ defined by (1.1). Therefore, it is easy to see that (𝐻𝑝𝑧𝑤,⊛) is also Banach algebra, and hence, 𝒟𝑓 is a bounded operator on 𝐻𝑝𝑧𝑤. Then it follows from (3.2) that 𝑊span𝑘𝑧𝑤𝑓∶𝑘≥0=span(𝑧𝑤)𝑘(𝑘!)2𝒟⊛𝑓(𝑧𝑤)∶𝑘≥0=span𝑓(𝑧𝑤)𝑘(𝑘!)2∶𝑘≥0=clos𝒟𝑓span(𝑧𝑤)𝑘∶𝑘≥0=clos𝒟𝑓𝐻𝑝𝑧𝑤.(3.3) Thus, 𝑓 is a cyclic vector for 𝑊𝑧𝑤 if and only if 𝒟𝑓 has a dense range. Let us show that the latter is equivalent to the condition 𝑓(0)≠0. Clearly, if 𝒟𝑓 has a dense range then 𝑓(0)≠0. Conversely, let 𝑓(0)≠0. We will prove actually more strong result that 𝒟𝑓 is invertible in 𝐻𝑝𝑧𝑤. Really, let us rewrite the operator 𝒟𝑓 in the form 𝒟𝑓=𝑓(0)𝐼+𝒦𝜕2𝑓/𝜕𝑧𝜕𝑤, where 𝐼 is the identity operator in 𝐻𝑝𝑧𝑤 and 𝒦𝜕2𝑓/𝜕𝑧𝜕𝑤𝑔(𝑧𝑤)=𝑧0𝑤0𝑓𝑧+((𝑧−𝑢)(𝑤−𝑣))(𝑧−𝑢)(𝑤−𝑣)𝑓𝑧𝑤𝑔((𝑧−𝑢)(𝑤−𝑣))(𝑢𝑣)𝑑𝑣𝑑𝑢.(3.4) Since 𝜕2𝑓/𝜕𝑧𝜕𝑤 is a continuous function, it is easy to see that 𝒦𝜕2𝑓/𝜕𝑧𝜕𝑤 is a compact operator (even Volterra operator) on 𝐻𝑝𝑧𝑤. Now, as in the proof of Theorem 2.5, it follows from Titchmarsh Convolution Theorem that ker𝒟𝑓={0}. Then, again by the Fredholm theorem we assert that 𝒟𝑓 is invertible, which completes the proof.

In conclusion, note that the study of the double integration operator 𝑊 in the Lebesgue space 𝐿2([0,1]×[0,1]) was originated by Donoghue, Jr., in [18]. He showed that the operator 𝑊 is not unicellular. Atzmon and Manos [19] proved that the multiplicity of spectrum 𝜇(𝑊) of the operator 𝑊 is equal to +∞ (we recall that the multiplicity of spectrum of the Banach space operator 𝐴∈ℒ(𝑋) is defined by 𝜇(𝐴)∶=min{card𝐸∶span{𝐴𝑛𝐸∶𝑛≥0}=𝑋}). Some related results for 𝑊 are also contained in the paper [15] by Karaev.