On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra

In the present paper we consider the shift operator S on the Wiener algebra W (D) of analytic functions on the unit disc D of the complex plane C. A complex number lambda is called an extended eigenvalue of S if there exists a nonzero operator A satisfying the equation AS = lambda SA. We prove that the set of all extended eigenvalues of S is precisely the set (D) over bar, and describe in terms of multiplication operators and composition operators the set of all corresponding extended eigenvectors of S. (C) 2009 Elsevier Ltd. All rights reserved.