Connections between Deddens Algebras and Extended Eigenvectors

A complex number $\lambda$ 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 $A\colon C_{A}(\mathbb{D}) \to C_{A}(\mathbb{D})$ satisfying the equation $AS=\lambda S\mspace{-3mu}A$. 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 $S$.