#### Abstract

Let be a Banach space of complex-valued functions that are continuous on , where is the unit disc in the complex plane , and have th partial derivatives in which can be extended to functions continuous on , and let denote the subspace of functions in which are analytic in (i.e., ). The double integration operator is defined in by the formula . By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator , where is an invariant subspace of , and study its properties. We also study invertibility of the elements in with respect to the Duhamel product.

#### 1. Introduction and Backgrounds

Let denote the Fréchet space of functions that are holomorphic in the bidisc . The product we define on this space is which obviously defines an integrodifferential operator . This product is a natural extension of the Duhamel product on [1]: where the integrals are taken over the segment joining the points 0 and .

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 , the Hardy space (which is the space of all holomorphic functions on the open unit disc for which the norm is finite) is a Banach algebra under the Duhamel product .

The Hardy space of the polydisc, , is defined as those functions analytic on for which the following norm is finite: If , this is a Banach space, and if , this is a Fréchet space [3]. In [3], Merryfield and Watson proved that for is a Banach algebra with respect to the product (1.1).

In the present paper we prove that the space 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 , where By using product (1.6) we also describe commutant of the operator , that is, the set of bounded linear operators on commuting with . Moreover, we describe the set of cyclic vectors of the double integration operator acting on the closed subspace We recall that a vector is called cyclic vector for the operator (Banach algebra of all bounded linear operators on a Banach space ) if where denotes the closure of the linear hull of the set .

#### 2. Description of

For any operator its commutant is defined by

The study of commutant of the concrete operator 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 on a Banach space (recall that a closed subspace is called hyperinvariant subspace for the operator , if it is invariant for any operator ). 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 on the closed subspace of the space . First, we prove the following lemma, which shows that is a Banach algebra under the Duhamel product given by formula (1.6).

Lemma 2.1. * is a Banach algebra.*

*Proof. *Indeed, let be two functions. The norm in is defined by
Using (1.6), (2.2) and the Leibnitz formula for the partial derivatives of the product , it can be proved (which is omitted) that (see e.g., the method of the paper [15, 16])
for some constant , which proves the lemma.

The main result of this section is the following theorem.

Theorem 2.2. *Let be an operator. Then if and only if there exists a function such that , where is the Duhamel operator defined by
*

*Proof. *Let , that is,
Then we have that
for all , whence by computing we have
or
for all .

From (2.8) we have by induction that
Indeed, for we have from (2.8) , as desired.

Assume for that
For we have from (2.8) that
Now, by considering (2.10) from the latter equality we have
which proves (2.9).

Now, let us show that
For this purpose, first show that
for all . Indeed, it follows directly from (1.6) that 1 is the unit with respect to the Duhamel product in , and for every . 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 , which means that
and hence
for all polynomials . Thus, by Lemma 2.1 and Weierstrass approximation theorem, we deduce that
where
Thus,
for all and some . Conversely, if , then commutes with . Since by Lemma 2.1 is a Banach algebra with respect to the Duhamel product , is bounded operator on . The theorem is proved.

Corollary 2.3. *For a function in there exists a unique commutant of the operator such that .*

Corollary 2.4. *One has , where stands for the bicommutant of the operator .*

*Proof. *It suffices to prove that for every , in . Indeed, by Theorem 2.2, there exist such that
for all , where
Since the usual convolution operators and are commuting operators, we have
which proves the corollary.

Theorem 2.5. *An operator will be an isomorphism of the space into itself and commutes with if and only if it can be written in the form
**
and.*

*Proof. *If is an isomorphism of the space into itself and commutes with , then by Theorem 2.2 we have for representation (2.26) with . Clearly, it follows from this equality and (2.26) that .

Conversely, suppose that has the form (2.26) with , and prove then that and is an isomorphism on . Indeed, the inclusion follows directly from Theorem 2.2. On the other hand, it is easy to see from the representation (2.26) that
where ,
is the usual convolution operator on . It is not difficult to see that is a compact operator on .

Let us show that . Indeed, let , where . Then, , that is,
By standard calculation, we obtain from (2.29) that
where , are constants. Since
we have that . On the other hand, since
for all , we obtain that . Thus,
for all and . Now, by considering that and is a continuous function on , by the Titchmarsh Convolution Theorem [17] for functions of several variables we deduce from (2.33) that for all , that is, . Since is compact, it follows from Fredholm alternative that is invertible in , that is, 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 belonging to and satisfying , there exists a unique isomorphism of the space such that commutes with and .*

Corollary 2.7. *If and , then the integrodifferential equation
**
has a unique solution for any right-hand side .*

Corollary 2.8. *The unique maximal ideal of the Banach algebra is ; that is, the maximal ideal space of consists of one homomorphizm, namely evaluation at the origin .*

#### 3. Cyclic Vectors of

Let us consider the restricted operator . 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 . Then
**
if and only if .*

*Proof. *It is easy to verify that
for all . Let us define the integrodifferential operator (or, briefly, the Duhamel operator) defined by . By the known result of Merryfield and Watson (see [3, Corollary 2.6]), , is the Banach algebra with respect to the Duhamel product defined by (1.1). Therefore, it is easy to see that is also Banach algebra, and hence, is a bounded operator on . Then it follows from (3.2) that
Thus, is a cyclic vector for if and only if has a dense range. Let us show that the latter is equivalent to the condition . Clearly, if has a dense range then . Conversely, let . We will prove actually more strong result that is invertible in . Really, let us rewrite the operator in the form , where is the identity operator in and
Since is a continuous function, it is easy to see that is a compact operator (even Volterra operator) on . Now, as in the proof of Theorem 2.5, it follows from Titchmarsh Convolution Theorem that . Then, again by the Fredholm theorem we assert that is invertible, which completes the proof.

In conclusion, note that the study of the double integration operator in the Lebesgue space was originated by Donoghue, Jr., in [18]. He showed that the operator is not unicellular. Atzmon and Manos [19] proved that the multiplicity of spectrum of the operator is equal to (we recall that the multiplicity of spectrum of the Banach space operator is defined by ). Some related results for are also contained in the paper [15] by Karaev.