Multivariable Nehari problem and interpolation

Abstract

In this paper we obtain a multivariable Nehari type theorem for Hankel operators, extending the classical result as well as Treil–Volberg generalization. The result is based on a new generalization of the noncommutative commutant lifting theorem which extends to several variables the commutant lifting result obtained by Treil and Volberg, and recently generalized by Biswas, Foiaş, and Frazho. An extension of I.S. Iokhvidov–Ky Fan theorem is obtained and used to prove a multivariable Adamian–Arov–Krein type theorem for generalized Hankel operators. As consequences, we obtain Carathéodory and Nevanlinna–Pick type interpolation results for “meromorphic” operators on Fock spaces (resp. operator-valued meromorphic functions on the unit ball of C n ). The Nehari type results are used to obtain descriptions of Hankel operators acting on Fock spaces or weighted Fock spaces (multivariable Dirichlet type spaces) and to solve new “weighted” interpolation problems (e.g. Sarason) for noncommutative analytic Toeplitz algebras, which have consequences to the operator-valued analytic interpolation in the unit ball of C n . Moreover, using central intertwining liftings, we obtain explicit formulas for certain solutions of the weighted interpolation problems. As another application of the generalized noncommutative commutant lifting theorem, we derive an abstract interpolation problem for multipliers. This is used to solve a left tangential Nevanlinna–Pick type interpolation problem with operatorial argument for noncommutative analytic Toeplitz algebras.