INTERTWINING RELATIONS AND EXTENDED EIGENVALUES FOR ANALYTIC TOEPLITZ OPERATORS

We study the intertwining relation XT phi = T psi X where T-phi and T-psi, are the Toeplitz operators induced on the Hardy space H-2 by analytic functions phi and psi, bounded on the open unit disc U, and X is a nonzero bounded linear operator on H-2. Our work centers on the connection between intertwining and the image containment psi(U) subset of phi(U), as well as on the nature of the intertwining operator X. We use our results to study the "extended eigenvalues" of analytic Toeplitz operators T-phi, i.e., the special case XT lambda phi = T phi X, where lambda is a complex number.