Connections between Deddens algebras and extended eigenvectors

A complex number. is called an extended eigenvalue of the shift operator S, Sf = zf, on the disc algebra C(A)(D) if there exists a nonzero operator A: C(A)(D) -> C(A)(D) satisfying the equation AS = lambda SA. 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.