On some operator equations in the space of analytic functions and related questions

We investigate extended eigenvalues, extended eigenvectors, and cyclicity problems for some convolution operators. By using the Duhamel product technique, we also estimate the norm of the inner derivation operator ∆A.


INTRODUCTION AND BACKGROUND
Let B (E) be an algebra of all continuous linear operators acting on the topological vector space E. The operator equation AX = XB (1) naturally arises in numerous issues of spectral theory of operators, representation theory, stability theory (Lyapunov's equation), etc.For example, if the set of solutions of Eq. ( 1) contains a boundedly invertible operator X 0 , then A and B are similar, B = X −1 0 AX 0 , and hence have many common spectral properties.In general case, it is of interest to describe the set of all solutions of Eq. (1).
If B = λ A, λ ∈ C, then following [1], one refers to λ as an extended eigenvalue of A, and each bounded solution X of the equation AX = λ XA, i.e., Eq. ( 1) with B = λ A, is called an extended eigenvector of A.
In this paper we investigate the so-called extended eigenvalues and extended eigenvectors and cyclicity problems for some convolution operators acting on the space of analytic functions defined on the starlike domain D of the complex plane.Our investigation is motivated by the results of Nagnibida's paper [11].By using the Duhamel product method (see [13]), we also give a lower estimate for the inner derivation operator The set of intertwining operators for the pair V β , λV β with β > 0 and λ ∈ C was studied by Malamud in [3,9,10].Namely, he showed that there exists a nonzero intertwining operator for the pair V β , λV β only if λ > 0. Furthermore, the paper [10] provides a description of the set V β λ of all intertwining operators for the pair V β , λV β for λ > 0. For β = 1, the latter result was reproved by another method by Biswas, Lambert, and Petrovic [1], and Karaev [6].For more details, see [1,2,4,5,9,10].
Let α be a fixed complex number, let D be a simply connected region in the complex plane C that is starlike with respect to the point z = α (i.e., λ z + (1 − λ ) α ∈ D), and let A (D) be the space of all singlevalued and analytic functions in D that have a topology of uniform convergence on compact subsets.It is well known that A (D) is a Frechet space.By J α we shall denote the integration operator in the space A (D) defined by the formula where the integration is performed over straight-line segments connecting the points α and z (z ∈ A (D)).
Recall that for f , g ∈ A (D) their α-Duhamel product is defined by where the integrals are taken over the segment joining the points α and z (z ∈ A (D)) .It is easy to see that the α-Duhamel product satisfies all the axioms of multiplication, A (D) is an algebra with respect to α as well, and the function f (z) ≡ 1 is the unit element of the algebra is called the α-Duhamel operator on A (D) .

EXTENDED EIGENVALUES AND EXTENDED EIGENVECTORS FOR SOME CONVOLUTION OPERATORS
Let D ⊂ C be a starlike region with respect to the origin.For any fixed nonzero function f ∈ A (D) , let K f be the usual convolution operator acting on the space A (D) by the formula It follows from the classical Titchmarsh convolution theorem and uniqueness theorem for analytic functions that ker (K f ) = {0} .This means that 0 is not an extended eigenvalue of the operator K f , and therefore ext (K f ) ⊂ C\ {0} (here ext (K f ) denotes the set of all extended eigenvalues of the operator K f ).
The integration operator J on A (D) is defined by n≥0 is a complete system in A (D) , we will denote by Λ f the set of all λ ∈ C\ {0} for which the diagonal operator The following theorem gives necessary and sufficient conditions under which the set Λ f lies in the set ext (K f ) .Our result is apparently the first result in the "extended theory" for more general convolution operators, which is an extension of Karaev's result [7, Theorem 2, (ii)].The related results for the integration operator are considered in [4,7].
Theorem 1.Let f ∈ A (D) be a nonzero function.Suppose that the system (J f ) n n≥0 is complete in A (D) .Let A ∈ B (A (D)) be a nonzero operator and λ ∈ Λ f be any number.Then Proof.By using the usual Duhamel product , which is defined by we have that any function f 1 ∈ A (D) defines the continuous operator (Duhamel operator) Then we have Now, let λ ∈ Λ f be any number, and suppose that Then, obviously λ n K n f Ag = AK n f g for all g ∈ A (D) and n ≥ 0. In particular, putting g = 1 in the last equality, we have for all n ≥ 0. Since K f = D J f , clearly we have for all n ≥ 0. This shows that for all n ≥ 0. Since (J f ) n n≥0 is a complete system of the space A (D) and D {λ } is a continuous operator on A (D) , it follows from the last equalities that Conversely, let us show that if A has the form A = D ϕ D {λ } , where ϕ ∈ A (D) , it satisfies the equation AK f = λ K f A. In fact, by considering the formula AK f = D J f , and commutativity of the product , we obtain for all n ≥ 0. By considering completeness of the system (J f ) n n≥0 in A (D) , from the last equalities we deduce that AK f = λ K f A. The theorem is proved.

CYCLIC VECTORS OF CONVOLUTION OPERATOR K f ,α
Let D be a starlike region in the complex plane C with respect to z = α.Our next result describes all cyclic vectors of some convolution operators of the form Theorem 2. Let f ∈ A (D) , and assume that (J α f ) α n n≥0 is a complete system in A (D) .If g ∈ A (D) , then g is a cyclic vector for the convolution operator K f ,α if and only if g (α) = 0.
Proof.It follows from the definition of α-Duhamel product Then according to the condition of the theorem, we obtain that Now, if g (α) = 0, then by virtue of Nagnibida's result operator D g,α is invertible in A (D) , which implies that D g A (D) = A (D) .
Hence E g = A (D) , which shows that g is a cyclic vector for the convolution operator K f ,α .
where F := f G. Clearly, F = F A,X .The equality (6) shows that 1 ∈ σ p (D F (AX − XA)) , that is, 1 is the eigenvalue of the operator D F ∆ A (X) .Therefore, By taking supremum over the operators X with X ≤ 1, we have from this inequality that sup

By denoting C
> 0, we have the desired result.The theorem is proved.