Intertwining relations and extended eigenvalues for analytic Toeplitz operators

Abstract: We study the intertwining relation XT_f = T_gX, where T_f and T_g are the Toeplitz operators induced on the Hardy space H² by analytic functions f and g that are bounded on the open unit disc U. Our work centers on the connection between intertwining and the containment of the image of g in the image of f, as well as on the nature of the intertwining operator X. We use our results to study the "extended eigenvalues" of analytic Toeplitz operators.