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

Abstract

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 λ is called an extended eigenvalue of S if there exists a nonzero operator A satisfying the equation A S = λ S A . We prove that the set of all extended eigenvalues of S is precisely the set D ¯ , and describe in terms of multiplication operators and composition operators the set of all corresponding extended eigenvectors of S .