Extended eigenvalues for bilateral weighted shifts

A complex scalar λ is said to be an extended eigenvalue for an operator A on a Hilbert space H if there is a non-zero operator X such that AX=λXA, and in that case, X is said to be an extended eigenoperator. It is shown that if a bilateral weighted shift has a non-unimodular extended eigenvalue then every extended eigenoperator for A is strictly lower triangular. Also, it is shown that the set of the extended eigenvalues for an injective bilateral weighted shift is either C∖D or C∖{0} or D‾∖{0}, or T, and some examples are constructed in order to show that each of the four shapes does happen. Further, it is shown that the set of the extended eigenvalues for an injective bilateral weighted shift with an even sequence of weights is either C\{0} or T, and that the set of the extended eigenvalues for an invertible bilateral weighted shift is T. Finally, a factorization result is provided for the extended eigenoperators corresponding to a unimodular extended eigenvalue of an injective bilateral weighted shift.