Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)W(D) of analytic functions on the unit disc DD of the complex plane CC. A complex number λλ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVAAV=λVA. We prove that the set of all extended eigenvalues of V   is precisely the set C⧹{0}C⧹{0}, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of VV. The similar result for some weighted shift operator on ℓpℓp spaces is also obtained.