On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra (Q1036827)

A complex number \(\lambda\) is an extended eigenvalue of a bounded operator \(X\) on a Banach space if there is a nonzero bounded operator \(Y\) such that \(XY = \lambda YX\). In this case, \(Y\) is said to be an extended eigenvector associated with the eigenvalue \(\lambda\). A function \( f(z) = \sum_{n=0}^{\infty} a_n z^n\) is in the Wiener algebra \(W(\mathbb D)\) if it is analytic on the unit disk \(\mathbb D\) and \( f(z) = \sum_{n=0}^{\infty} | a_n | < \infty\). The author's abstract and main result claim that the set of extended eigenvalues of the shift operator \(S=M_z\) on \(W(\mathbb D)\) is equal to the closed unit disk \(\bar{\mathbb D }\). This is actually not the case. In fact, the author correctly observes prior to the main result that \(\lambda =0\) cannot be an extended eigenvalue of \(S\). The author's result is essentially the observation that if \(\lambda \in \bar{\mathbb D} \setminus \{0\}\), \(g\) is any nonzero function in the multiplier algebra of \(W(\mathbb D)\), and \(C_\lambda\) denotes the composition operator \(C_\lambda(f) = f(\lambda z)\), then \(A=M_gC_\lambda\) is an extended eigenvector for \(S\) associated with \(\lambda\).