Connections between Deddens algebras and extended eigenvectors

Abstract

A complex number λ is called an extended eigenvalue of the shift operator S, Sf = zf, on the disc algebra \(C_A (\mathbb{D})\) if there exists a nonzero operator \(A:{\text{ }}C_A (\mathbb{D}) \to C_A (\mathbb{D})\) satisfying the equation AS = λSA. We describe the set of all extended eigenvectors of S in terms of multiplication operators and composition operators. It is shown that there are connections between the Deddens algebra associated with S and the extended eigenvectors of S.