Intertwining Relations and Extended Eigenvalues for Analytic Toeplitz Operators

We study the intertwining relation XTφ = TψX where Tφ and Tψ are the Toeplitz operators induced on the Hardy space H 2 by analytic functions φ and ψ, bounded on the open unit disc U, and X is a nonzero bounded linear operator on H. Our work centers on the connection between intertwining and the image containment ψ(U) ⊂ φ(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 Tφ, i.e., the special case XTλφ = TφX, where λ is a complex number.