On Some Applications of a Special Integrodifferential Operators

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 n A C n A D × D denote the subspace of functions in C n D × D which are analytic in D × D i.e., C n A C n D × D ∩ Hol D × D . The double integration operator is defined in C n A by the formula Wf z,w ∫z 0 ∫w 0 f u, v dv du. 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 ∈ C n A : f z,w f zw } is an invariant subspace of W , and study its properties. We also study invertibility of the elements in C n A with respect to the Duhamel product.

f zw } is an invariant subspace of W, and study its properties.We also study invertibility of the elements in C n A with respect to the Duhamel product.

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| < 1 and |w| < 1}.The product we define on this space is f g z, w : 1.1 which obviously defines an integrodifferential operator D f , D f g : f g.This product is a natural extension of the Duhamel product on Hol D 1 : 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 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: 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 ≥ 1 H 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 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 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, A 2 x, . . .X, 1.8 where span{x, Ax, A 2 x, . ..} denotes the closure of the linear hull of the set {x, Ax, A 2 x, . ..}.

Description of {W zw }
For any operator A ∈ L X its commutant {A} is defined by 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 n A .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 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 for some constant C n > 0, which proves the lemma.
The main result of this section is the following theorem.
Proof.Let T ∈ {W zw } , that is, Then we have that for all k 0, 1, . .., whence by computing W zw zw k we have or for all k 0, 1, . ... From 2.8 we have by induction that Indeed, for k 1 we have from 2.8 T zw W zw T 1, as desired.Assume for k n that

2.10
For k n 1 we have from 2.8 that Now, by considering 2.10 from the latter equality we have 2.12 which proves 2.9 .Now, let us show that

2.13
For this purpose, first show that 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 zw f zw f zw for every f ∈ E zw .From this by induction we have equality 2.14 we omit details .
Then we have which proves 2.13 .Now by combining 2.9 and 2.13 we have for all k ≥ 0, which means that

and hence
Tp zw p zw T 1, 2.18 for all polynomials p.Thus, by Lemma 2.1 and Weierstrass approximation theorem, we deduce that Proof.It suffices to prove that T 1 T 2 T 2 T 1 for every T 1 , T 2 in {W zw } .Indeed, by Theorem 2.2, there exist ϕ, ψ ∈ E zw such that

Journal of Function Spaces and Applications
for all f ∈ E zw , where

2.24
Since the usual convolution operators K Φ and K Ψ are commuting operators, we have 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
26 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 zw T .Clearly, it follows from this equality and 2.26 that ϕ 0 T 1 | zw 0 / 0. Conversely, suppose that T has the form 2.26 with ϕ 0 T 1 | 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 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 ker By standard calculation, we obtain from 2.29 that where c 1 , c 2 are constants.Since we have that c 2 0. On the other hand, since for all z ∈ D, we obtain that c 1 0. Thus, 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, ker T {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 T 1 ϕ 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 .

Cyclic Vectors of W zw
Let us consider the restricted operator W zw W | H p zw .In this section we will describe the set of all cyclic vectors of this operator.The main result of this section is the following.

Journal of Function Spaces and Applications
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

3.4
Since ∂ 2 f/∂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 p zw .Now, as in the proof of Theorem 2.5, it follows from Titchmarsh Convolution Theorem that ker D 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 L 2 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{card E : span{A n E : n ≥ 0} X} .Some related results for W are also contained in the paper 15 by Karaev.
D denote the subspace of functions in C n D × D which are analytic in D × D i.e., C n A C n D × D ∩ Hol D × D .The double integration operator is defined in C n A by the formula Wf z, w z 0 w 0 f u, v dv du.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 n A : f z, w

Corollary 2 . 7 .
If ϕ ∈ E zw and ϕ 0 / 0, then the integrodifferential equation ϕ − u w − v z − u w − v ϕ zw z − u w − v x uv dv du y zw 2.34 has a unique solution for any right-hand side y ∈ E zw .
p zw .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 p zw and K ∂ 2 f/∂z∂w g zw z 0 w 0 f z z − u w − v z − u w − v f zw z − u w − v g uv dv du.
By the known result of Merryfield and Watson see 3, Corollary 2.6 , H p D 2 , p ≥ 1, is the Banach algebra with respect to the Duhamel product defined by 1.1 .Therefore, it is easy to see that H p zw .