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) of analytic functions on the unit disc B of the complex plane C. A complex number A is called an extended eigenvalue of V if there exists a nonzero operator A satisfying the equation A V = lambda VA. We prove that the set of all extended eigenvalues of V is precisely the set C\{0}, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V. The similar result for some weighted shift operator on l(p) spaces is also obtained. (C) 2008 Elsevier GmbH. All rights reserved.