Intertwining relations and extended eigenvalues for analytic Toeplitz operators

Abstract

We study the intertwining relation XT_φ=T_ψX where T_φ and T_ψ are the Toeplitz operators induced on the Hardy space H² by analytic functions φ and ψ, bounded on the open unit disc , and X is a nonzero bounded linear operator on H². Our work centers on the connection between intertwining and the image containment , 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_φ, i.e., the special case XT_λφ=T_φX, where λ is a complex number.