On Some Applications of a Special Integrodifferential Operators

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 𝐶^𝐴(𝑛)=𝐶^𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐶^(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐶^𝐴(𝑛)=𝐶^(𝑛)(𝔻×𝔻)∩ℋ𝑜𝑙(𝔻×𝔻)). The double integration operator is defined in 𝐶^𝐴(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∣𝐸_𝑧𝑤, where 𝐸_𝑧𝑤={𝑓∈𝐶^𝐴(𝑛)∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤)} is an invariant subspace of 𝑊, and study its properties. We also study invertibility of the elements in 𝐶^𝐴(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𝜕𝑤^𝑧0^𝑤0𝑓(𝑧−𝑢,𝑤−𝑣)𝑔(𝑢,𝑣)𝑑𝑣𝑑𝑢,(1.1) which obviously defines an integrodifferential operator 𝒟_𝑓,𝒟_𝑓𝑔∶=𝑓⊛𝑔. This product is a natural extension of the Duhamel product on ℋ𝑜𝑙(𝔻) [1]: (𝑓⊛𝑔)(𝑧)∶=𝑑𝑑𝑧^z₀𝑓(𝑧−𝑡)𝑔(𝑡)𝑑𝑡=^z₀𝑓^(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 𝐻^𝑝(𝔻) (which is the space of all holomorphic functions on the open unit disc 𝔻 for which the norm ‖𝑓‖_𝐻^𝑝=sup_0<𝑟<121𝜋^02𝜋||𝑓𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^1/𝑝(1.3) is finite) is a Banach algebra under the Duhamel product ⊛.

The Hardy space of the polydisc, 𝐻^𝑝(𝔻^𝑛), is defined as those functions analytic on 𝔻^𝑛∶=𝔻×⋯×𝔻 for which the following norm is finite: ‖𝑓‖_𝐻^𝑝∶=sup_𝑟1<1⋯sup_𝑟𝑛<1(21𝜋)^𝑛^02𝜋⋯^02𝜋||𝑓𝑟₁𝑒^𝑖𝜃1,…,𝑟_𝑛𝑒^𝑖𝜃𝑛||^𝑝𝑑𝜃₁⋯𝑑𝜃_𝑛^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≥1𝐻^𝑝(𝔻^𝑛) is a Banach algebra with respect to the product (1.1).

In the present paper we prove that the space 𝐸_𝑧𝑤 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 (𝐸_𝑧𝑤,⊛), where (𝑓⊛𝑔)(𝑧𝐸𝑤_𝑧𝑤)=∶=𝜕₂𝑓∈𝐶^𝐴(𝑛)∶𝑓(𝑧,𝑤)=𝑓(𝑧𝑤),(1.5)𝜕𝑧𝜕𝑤^𝑧0^𝑤0𝑓((𝑧−𝑢)(𝑤−𝑣))𝑔(𝑢𝑣)𝑑𝑣𝑑𝑢.(1.6) By using product (1.6) we also describe commutant of the operator 𝑊_𝑧𝑤∶=𝑊∣𝐸_𝑧𝑤, that is, the set of bounded linear operators on 𝐸_𝑧𝑤 commuting with 𝑊_𝑧𝑤. Moreover, we describe the set of cyclic vectors of the double integration operator 𝑊_𝑧𝑤 acting on the closed subspace 𝐻^𝑝𝑧𝑤∶=𝑓(𝑧,𝑤)∈𝐻^𝑝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 span𝑥,𝐴𝑥,𝐴²𝑥,…=𝑋,(1.8) where span{𝑥,𝐴𝑥,𝐴²𝑥,…} denotes the closure of the linear hull of the set {𝑥,𝐴𝑥,𝐴²𝑥,…}.

2. Description of {𝑊_𝑧𝑤}^

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 𝑊_𝑧𝑤 on the closed subspace 𝐸_𝑧𝑤 of the space 𝐶^𝐴(𝑛). First, we prove the following lemma, which shows that 𝐸_𝑧𝑤 is a Banach algebra under the Duhamel product ⊛ given by formula (1.6).

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

Proof. Indeed, let 𝑓,𝑔∈𝐸_𝑧𝑤 be two functions. The norm in 𝐸_𝑧𝑤 is defined by ‖𝑓‖_𝑛∶=maxmax_{(𝑧,𝑤)∈𝔻2}||||𝜕^|𝛼|𝑓(𝑧𝑤)𝜕𝑧^𝛼1𝜕𝑤^𝛼2||||∶|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≤𝐶_𝑛‖𝑓‖_𝑛‖𝑔‖_𝑛(2.3) for some constant 𝐶_𝑛>0, which proves the lemma.

The main result of this section is the following theorem.

Theorem 2.2. Let 𝑇∈ℒ(𝐸_𝑧𝑤) be an operator. Then 𝑇∈{𝑊_𝑧𝑤}^ if and only if there exists a function 𝜑∈𝐸_𝑧𝑤 such that 𝑇=𝒟_𝜑, where 𝒟_𝜑 is the Duhamel operator defined by 𝒟_𝜑𝑓(𝑧𝑤)=(𝜑⊛𝑓)(𝑧𝑤)=𝜕²𝜕𝑧𝜕𝑤^𝑧0^𝑤0𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢.(2.4)

Proof. Let 𝑇∈{𝑊_𝑧𝑤}^, that is, 𝑇𝑊_𝑧𝑤=𝑊_𝑧𝑤𝑇(2.5) Then we have that 𝑇𝑊_𝑧𝑤(𝑧𝑤)^𝑘=𝑊_𝑧𝑤𝑇(𝑧𝑤)^𝑘,(2.6) for all 𝑘=0,1,…, whence by computing 𝑊_𝑧𝑤(𝑧𝑤)^𝑘 we have 𝑇^𝑧0^𝑤0(𝑢𝑣)^𝑘𝑑𝑣𝑑𝑢=𝑇^𝑧0𝑢^𝑘^𝑤0𝑣^𝑘𝑑𝑣𝑑𝑢=𝑇^𝑧0𝑢^𝑘𝑤^𝑘+1=𝑇𝑧_𝑘+1𝑘+1𝑑𝑢𝑤^𝑘+1(𝑘+1)²=(𝑘+11)²𝑇(𝑧𝑤)^𝑘+1,(2.7) or 𝑇(𝑧𝑤)^𝑘+1=(𝑘+1)²𝑊_𝑧𝑤𝑇(𝑧𝑤)^𝑘,(2.8) for all 𝑘=0,1,….
From (2.8) we have by induction that 𝑇(𝑧𝑤)^𝑘=𝑊^𝑘𝑧𝑤𝑇1^𝑘_𝑠=1𝑠²(𝑘=1,2,…).(2.9) Indeed, for 𝑘=1 we have from (2.8) 𝑇(𝑧𝑤)=𝑊_𝑧𝑤𝑇1, as desired.
Assume for 𝑘=𝑛 that 𝑇(𝑧𝑤)^𝑛=𝑊^𝑛𝑧𝑤𝑇1^𝑛_𝑠=1𝑠².(2.10) For 𝑘=𝑛+1 we have from (2.8) that 𝑇(𝑧𝑤)^𝑛+1=(𝑛+1)²𝑊_𝑧𝑤𝑇(𝑧𝑤)^𝑛.(2.11) Now, by considering (2.10) from the latter equality we have 𝑇(𝑧𝑤)^𝑛+1=(𝑛+1)²𝑊_𝑧𝑤𝑊^𝑛𝑧𝑤𝑇1^𝑛_𝑠=1𝑠²=𝑊_{𝑛+1𝑧𝑤}𝑇1(𝑛+1)^2𝑛_𝑠=1𝑠²=𝑊^{𝑛+1𝑧𝑤}𝑇1^𝑛+1_𝑠=1𝑠²,(2.12) which proves (2.9).
Now, let us show that 𝑊^𝑘𝑧𝑤𝑓(𝑧𝑤)=^𝑧0^𝑤0[(𝑧−𝑢)(𝑤−𝑣)]^𝑘−1[(𝑘−1)!]²𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢.(2.13) For this purpose, first show that 𝑊^𝑘𝑧𝑤𝑓(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 𝐸_𝑧𝑤, and 𝑊_𝑧𝑤𝑓=𝑧𝑤⊛𝑓(𝑧𝑤) for every 𝑓∈𝐸_𝑧𝑤. From this by induction we have equality (2.14) (we omit details).
Then we have 𝑊^𝑘𝑧𝑤𝑓(𝑧𝑤)=(𝑧𝑤)^𝑘[𝑘!]²⊛𝑓(𝑧𝑤)=𝜕²𝜕𝑧𝜕𝑤^𝑧0^𝑤0[(𝑧−𝑢)(𝑤−𝑣)]^𝑘[(𝑘!)]²=1𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢(𝑘!)²^𝑧0^𝑤0=𝑘𝑘²[(𝑧−𝑢)(𝑤−𝑣)]^𝑘−1𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢₂𝑘²[(𝑘−1)!]²^𝑧0^𝑤0=[(𝑧−𝑢)(𝑤−𝑣)]^𝑘−1𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢_𝑧0^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𝜕𝑤^𝑧0^𝑤0=𝜕𝑓((𝑧−𝑢)(𝑤−𝑣))(𝑇1)(𝑢𝑣)𝑑𝑣𝑑𝑢₂𝜕𝑧𝜕𝑤^𝑧0^𝑤0=(𝑇1)((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢_𝑧0^𝑤0𝜕²=𝜕𝑧𝜕𝑤(𝑇1)((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢_𝑧0^𝑤0(𝑧−𝑢)(𝑤−𝑣)(𝑇1)_𝑧𝑤+(𝑇1)_𝑤((𝑧−𝑢)(𝑤−𝑣))=((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢+(𝑇1)(0)𝑓(𝑧𝑤)_𝑧0^𝑤0𝜑_𝑤+𝜑₀((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑_𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑓(𝑧𝑤),(2.19) where 𝜑∶=𝑇1∈𝐸_𝑧𝑤.(2.20) Thus, (𝑇𝑓)(𝑧𝑤)=𝜑(𝑧𝑤)⊛𝑓(𝑧𝑤)=𝒟_𝜑𝑓(𝑧𝑤)(2.21) for all 𝑓∈𝐸_𝑧𝑤 and some 𝜑∈𝐸_𝑧𝑤. Conversely, if 𝜑∈𝐸_𝑧𝑤, then 𝒟_𝜑 commutes with 𝑊_𝑧𝑤. Since by Lemma 2.1𝐸_𝑧𝑤 is a Banach algebra with respect to the Duhamel product ⊛, 𝒟_𝜑 is bounded operator on 𝐸_𝑧𝑤. The theorem is proved.

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

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

Proof. It suffices to prove that 𝑇₁𝑇₂=𝑇₂𝑇₁ for every 𝑇₁, 𝑇₂ in {𝑊_𝑧𝑤}^. Indeed, by Theorem 2.2, there exist 𝜑,𝜓∈𝐸_𝑧𝑤 such that 𝑇₁𝑓(𝑧𝑤)=𝜑(0)𝑓(𝑧𝑤)+^𝑧0^𝑤0+𝜑(_𝑤𝑧(−(𝑧𝑢−)(𝑢𝑤)(−𝑤𝑣−)𝜑𝑣)_𝑧𝑤)=𝜑(0)𝐼+𝒦_Φ((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑇𝑓(𝑧𝑤),(2.22)₂𝑓(𝑧𝑤)=𝜓(0)𝑓(𝑧𝑤)+^𝑧0^𝑤0𝜓_𝑤+(𝑧(−(𝑧𝑢−)(𝑤𝑢)−(𝑤𝑣−)𝜓𝑣)_𝑧𝑤)=𝜓(0)𝐼+𝒦_Ψ((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢𝑓(𝑧𝑤)(2.23) for all 𝑓∈𝐸_𝑧𝑤, where Φ(𝑧𝑤)∶=𝜕²Ψ(𝑧𝑤)∶=𝜕𝑧𝜕𝜕₂𝑤𝜑(𝑧𝑤),𝒦𝜕𝑧𝜕𝑤𝜓(𝑧𝑤),_Φ𝑓(𝑧𝑤)=(Φ∗𝑓)(𝑧𝑤)∶=^𝑧0^𝑤0𝜕²=𝜕𝑧𝜕𝑤𝜑((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢_𝑧0^𝑤0𝜑_𝑤𝒦((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑_𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢,_Ψ𝑓(𝑧𝑤)=(Ψ∗𝑓)(𝑧𝑤)∶=^𝑧0^𝑤0𝜕²=𝜕𝑧𝜕𝑤𝜓((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢_𝑧0^𝑤0𝜓_𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜓_𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢.(2.24) Since the usual convolution operators 𝒦_Φ and 𝒦_Ψ are commuting operators, we have 𝑇₁𝑇₂=𝜑(0)𝐼+𝒦_Φ𝜓(0)𝐼+𝒦_Ψ=𝜓(0)𝐼+𝒦_Ψ𝜑(0)𝐼+𝒦_Φ=𝑇₂𝑇₁,(2.25) which proves the corollary.

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

Proof. If 𝑇∈ℒ(𝐸_𝑧𝑤) is an isomorphism of the space 𝐸_𝑧𝑤 into itself and commutes with 𝑊_𝑧𝑤, then by Theorem 2.2 we have for 𝑇 representation (2.26) with 𝑇𝑊_𝑧𝑤=𝑊_𝑧𝑤𝑇. 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∈{𝑊_𝑧𝑤}^ and 𝑇 is an isomorphism on 𝐸_𝑧𝑤. Indeed, the inclusion 𝑇∈{𝑊_𝑧𝑤}^ follows directly from Theorem 2.2. On the other hand, it is easy to see from the representation (2.26) that 𝑇=𝒟_𝜑=𝜑(0)𝐼+𝒦_Φ,(2.27) where 𝒦_Φ, 𝒦_Φ𝑓=^𝑧0^𝑤0𝜑_𝑤((𝑧−𝑢)(𝑤−𝑣))+(𝑧−𝑢)(𝑤−𝑣)𝜑_𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑓(𝑢𝑣)𝑑𝑣𝑑𝑢,(2.28) is the usual convolution operator on 𝐸_𝑧𝑤. It is not difficult to see that 𝒦_Φ is a compact operator on 𝐸_𝑧𝑤.
Let us show that ker𝒟_𝜑={0}. Indeed, let 𝒟_𝜑𝑓=0, where 𝑓∈𝐸_𝑧𝑤. Then, (𝜑(0)𝐼+𝒦_Φ)𝑓=0, that is, 𝜕²𝜕𝑧𝜕𝑤^𝑧0^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)𝑑𝑣𝑑𝑢||||_𝑧=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)𝑑𝑣𝑑𝑢||||_𝑤=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 𝒦_Φ is compact, it follows from Fredholm alternative that 𝑇 is invertible in 𝐸_𝑧𝑤, 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 𝐸_𝑧𝑤 and satisfying 𝜑(0)≠0, there exists a unique isomorphism 𝑇 of the space 𝐸_𝑧𝑤 such that 𝑇 commutes with 𝑊_𝑧𝑤 and 𝑇1=𝜑(𝑧𝑤).

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

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

3. Cyclic Vectors of 𝑊_𝑧𝑤

Let us consider the restricted operator 𝑊_𝑧𝑤=𝑊∣𝐻^𝑝𝑧𝑤. 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∈𝐻^𝑝𝑧𝑤. Then span𝑊^𝑛𝑧𝑤𝑓∶𝑛=0,1,2,…=𝐻^𝑝𝑧𝑤,(3.1) if and only if 𝑓∣_𝑧𝑤=0≠0.

Proof. It is easy to verify that 𝑊^𝑘𝑧𝑤𝑔(𝑧𝑤)=(𝑧𝑤)^𝑘(𝑘!)²⊛𝑔(𝑧𝑤)(𝑘≥0),(3.2) for all 𝑔∈𝐻^𝑝𝑧𝑤. Let us define the integrodifferential operator (or, briefly, the Duhamel operator) 𝒟_𝑓 defined by (𝒟_𝑓𝑔)(𝑧𝑤)=(𝑓⊛𝑔)(𝑧𝑤),𝑔∈𝐻^𝑝𝑧𝑤. By the known result of Merryfield and Watson (see [3, Corollary 2.6]), 𝐻^𝑝(𝔻²),𝑝≥1, is the Banach algebra with respect to the Duhamel product ⊛ defined by (1.1). Therefore, it is easy to see that (𝐻^𝑝𝑧𝑤,⊛) is also Banach algebra, and hence, 𝒟_𝑓 is a bounded operator on 𝐻^𝑝𝑧𝑤. Then it follows from (3.2) that span𝑊^𝑘𝑧𝑤𝑓∶𝑘≥0=span(𝑧𝑤)^𝑘(𝑘!)²=span𝒟_𝑓⊛𝑓(𝑧𝑤)∶𝑘≥0=clos𝒟_𝑓((𝑧𝑘𝑤!))^𝑘2∶𝑘≥0=clos𝒟_𝑓span(𝑧𝑤)^𝑘∶𝑘≥0𝐻^𝑝𝑧𝑤.(3.3) Thus, 𝑓 is a cyclic vector for 𝑊_𝑧𝑤 if and only if 𝒟_𝑓 has a dense range. Let us show that the latter is equivalent to the condition 𝑓(0)≠0. Clearly, if 𝒟_𝑓 has a dense range then 𝑓(0)≠0. Conversely, let 𝑓(0)≠0. We will prove actually more strong result that 𝒟_𝑓 is invertible in 𝐻^𝑝𝑧𝑤. Really, let us rewrite the operator 𝒟_𝑓 in the form 𝒟_𝑓=𝑓(0)𝐼+𝒦_𝜕²_{𝑓/𝜕𝑧𝜕𝑤}, where 𝐼 is the identity operator in 𝐻^𝑝𝑧𝑤 and 𝒦_𝜕²_{𝑓/𝜕𝑧𝜕𝑤}𝑔(𝑧𝑤)=^𝑧0^𝑤0𝑓_𝑧+((𝑧(𝑧−−𝑢𝑢))(𝑤(𝑤−−𝑣𝑣))𝑓)_𝑧𝑤((𝑧−𝑢)(𝑤−𝑣))𝑔(𝑢𝑣)𝑑𝑣𝑑𝑢.(3.4) Since 𝜕²𝑓/𝜕𝑧𝜕𝑤 is a continuous function, it is easy to see that 𝒦_𝜕²_{𝑓/𝜕𝑧𝜕𝑤} is a compact operator (even Volterra operator) on 𝐻^𝑝𝑧𝑤. Now, as in the proof of Theorem 2.5, it follows from Titchmarsh Convolution Theorem that ker𝒟_𝑓={0}. Then, again by the Fredholm theorem we assert that 𝒟_𝑓 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{𝐴^𝑛𝐸∶𝑛≥0}=𝑋}). Some related results for 𝑊 are also contained in the paper [15] by Karaev.