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)=∫0z∫0wf(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∂w∫0z∫0wf(z-u,w-v)g(u,v)dvdu,
which obviously defines an integrodifferential operator 𝒟f,𝒟fg:=f⊛g. This product is a natural extension of the Duhamel product on ℋol(𝔻) [1]:
(f⊛g)(z)∶=ddz∫0zf(z-t)g(t)dt=∫0zf′(z-t)g(t)dt+f(0)g(z),
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(12π∫02π|f(reiθ)|pdθ)1/p
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.
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)},(f⊛g)(zw)=∂2∂z∂w∫0z∫0wf((z-u)(w-v))g(uv)dvdu.
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
Hzwp:={f(z,w)∈Hp(D2):f(z,w)=f(zw)}.
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,
where span{x,Ax,A2x,…} denotes the closure of the linear hull of the set {x,Ax,A2x,…}.

2. Description of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M53"><mml:mo stretchy="false">{</mml:mo><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">}</mml:mo></mml:mrow><mml:mrow><mml:mi>′</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>

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

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∶=max{max(z,w)∈D2̅|∂|α|f(zw)∂zα1∂wα2|:|α|=α1+α2=0,1,…,n}.
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
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
Dφf(zw)=(φ⊛f)(zw)=∂2∂z∂w∫0z∫0wφ((z-u)(w-v))f(uv)dvdu.

Proof.

Let T∈{Wzw}′, that is,
TWzw=WzwT
Then we have that
TWzw(zw)k=WzwT(zw)k,
for all k=0,1,…, whence by computing Wzw(zw)k we have
T(∫0z∫0w(uv)kdvdu)=T(∫0zuk(∫0wvkdv)du)=T∫0zukwk+1k+1du=T(zk+1wk+1(k+1)2)=1(k+1)2T(zw)k+1,
or
T(zw)k+1=(k+1)2WzwT(zw)k,
for all k=0,1,….

From (2.8) we have by induction that
T(zw)k=WzwkT1∏s=1ks2(k=1,2,…).
Indeed, for k=1 we have from (2.8) T(zw)=WzwT1, as desired.

Assume for k=n that
T(zw)n=WzwnT1∏s=1ns2.
For k=n+1 we have from (2.8) that
T(zw)n+1=(n+1)2WzwT(zw)n.
Now, by considering (2.10) from the latter equality we have
T(zw)n+1=(n+1)2Wzw(WzwnT1∏s=1ns2)=Wzwn+1T1(n+1)2∏s=1ns2=Wzwn+1T1∏s=1n+1s2,
which proves (2.9).

Now, let us show that
(Wzwkf)(zw)=∫0z∫0w[(z-u)(w-v)]k-1[(k-1)!]2f(uv)dvdu.
For this purpose, first show that
(Wzwkf)(zw)=(zw)k[k!]2⊛f(zw),
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
(Wzwkf)(zw)=(zw)k[k!]2⊛f(zw)=∂2∂z∂w∫0z∫0w[(z-u)(w-v)]k[(k!)]2f(uv)dvdu=1(k!)2∫0z∫0wk2[(z-u)(w-v)]k-1f(uv)dvdu=k2k2[(k-1)!]2∫0z∫0w[(z-u)(w-v)]k-1f(uv)dvdu=∫0z∫0w[(z-u)(w-v)]k-1[(k-1)!]2f(uv)dvdu,
which proves (2.13).

Now by combining (2.9) and (2.13) we have
T(zw)k=∏s=1ks2∫0z∫0w[(z-u)(w-v)]k-1[(k-1)!]2T1dvdu,
for all k≥0, which means that
T(zw)k=(zw)k⊛T1(k≥0),
and hence
Tp(zw)=p(zw)⊛T1,
for all polynomials p. Thus, by Lemma 2.1 and Weierstrass approximation theorem, we deduce that
(Tf)(zw)=T1⊛f(zw)=∂2∂z∂w∫0z∫0wf((z-u)(w-v))(T1)(uv)dvdu=∂2∂z∂w∫0z∫0w(T1)((z-u)(w-v))f(uv)dvdu=∫0z∫0w∂2∂z∂w(T1)((z-u)(w-v))f(uv)dvdu=∫0z∫0w[(z-u)(w-v)(T1)zw((z-u)(w-v))+(T1)w((z-u)(w-v))]f(uv)dvdu+(T1)(0)f(zw)=∫0z∫0w[φw((z-u)(w-v))+(z-u)(w-v)φzw((z-u)(w-v))]f(uv)dvdu+φ0f(zw),
where
φ∶=T1∈Ezw.
Thus,
(Tf)(zw)=φ(zw)⊛f(zw)=Dφf(zw)
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)+∫0z∫0w[φw((z-u)(w-v))+(z-u)(w-v)φzw((z-u)(w-v))]f(uv)dvdu=(φ(0)I+KΦ)f(zw),(T2f)(zw)=ψ(0)f(zw)+∫0z∫0w[ψw((z-u)(w-v))+(z-u)(w-v)ψzw((z-u)(w-v))]f(uv)dvdu=(ψ(0)I+KΨ)f(zw)
for all f∈Ezw, where
Φ(zw)∶=∂2∂z∂wφ(zw),Ψ(zw)∶=∂2∂z∂wψ(zw),KΦf(zw)=(Φ*f)(zw)∶=∫0z∫0w∂2∂z∂wφ((z-u)(w-v))f(uv)dvdu=∫0z∫0w[φw((z-u)(w-v))+(z-u)(w-v)φzw((z-u)(w-v))]f(uv)dvdu,KΨf(zw)=(Ψ*f)(zw)∶=∫0z∫0w∂2∂z∂wψ((z-u)(w-v))f(uv)dvdu=∫0z∫0w[ψw((z-u)(w-v))+(z-u)(w-v)ψzw((z-u)(w-v))]f(uv)dvdu.
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,
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)+∫0z∫0w[φw((z-u)(w-v))+(z-u)(w-v)φzw((z-u)(w-v))]f(uv)dvdu,
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,
where 𝒦Φ,
KΦf=∫0z∫0w[φw((z-u)(w-v))+(z-u)(w-v)φzw((z-u)(w-v))]f(uv)dvdu,
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∂w∫0z∫0wφ((z-u)(w-v))f(uv)dvdu=0.
By standard calculation, we obtain from (2.29) that
∫0z∫0wφ((z-u)(w-v))f(uv)dvdu=c1z+c2,
where c1, c2 are constants. Since
0=(∫0z∫0wφ((z-u)(w-v))f(uv)dvdu)|z=0=(c1z+c2)|z=0=c2,
we have that c2=0. On the other hand, since
0=(∫0z∫0wφ((z-u)(w-v))f(uv)dvdu)|w=0=c1z,
for all z∈𝔻, we obtain that c1=0. Thus,
∫0z∫0wφ((z-u)(w-v))f(uv)dvdu=0,
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)+∫0z∫0w[φw((z-u)(w-v))+(z-u)(w-v)φzw((z-u)(w-v))]x(uv)dvdu=y(zw)
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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M207"><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>z</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

Let us consider the restricted operator Wzw=W∣Hzwp. 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∈Hzwp. Then
span{Wzwnf:n=0,1,2,…}=Hzwp,
if and only if f∣zw=0≠0.

Proof.

It is easy to verify that
Wzwkg(zw)=(zw)k(k!)2⊛g(zw)(k≥0),
for all g∈Hzwp. Let us define the integrodifferential operator (or, briefly, the Duhamel operator) 𝒟f defined by (𝒟fg)(zw)=(f⊛g)(zw),g∈Hzwp. 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 (Hzwp,⊛) is also Banach algebra, and hence, 𝒟f is a bounded operator on Hzwp. Then it follows from (3.2) that
span{Wzwkf:k≥0}=span{(zw)k(k!)2⊛f(zw):k≥0}=span{Df((zw)k(k!)2):k≥0}=closDfspan{(zw)k:k≥0}=closDfHzwp.
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 Hzwp. 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 Hzwp and
K∂2f/∂z∂wg(zw)=∫0z∫0w[fz((z-u)(w-v))+(z-u)(w-v)fzw((z-u)(w-v))]g(uv)dvdu.
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 Hzwp. 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{card E:span{AnE:n≥0}=X}). Some related results for W are also contained in the paper [15] by Karaev.

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