Interpolation and Commutant Lifting for Multipliers on Reproducing Kernel Hilbert Spaces

Abstract

We obtain an explicit representation formula and a Nevanlinna-Pick-type interpolation theorem for the multiplier space of the reproducing kernel space ℌ(k d ) of analytic functions on the d-dimensional complex unit ball with reproducing kernel k d (z, w) = 1/(1 — (z, w)). More generally, if k is a positive kernel on a set Ω such that 1/k has 1 positive square, then there is an embedding b of Ω into the ball B d (where d is the number of negative squares of 1/k) such that any multiplier W for ℌ (k) lifts to a multiplier F for the space ℌ (kd) on the ball (W = F o b). As a corollary, multipliers for ℌ (k) also have explicit realization formulas and satisfy a Nevanlinna-Pick-type interpolation theorem. All the results in fact extend to the case of matrix-and operator-valued multipliers and left tangential Nevanlinna-Pick interpolation. Contractive multiplier solutions of a given set of interpolation conditions correspond to unitary extensions of a partially defined isometric operator; hence a technique of Arov and Grossman can be used to give a linear-fractional parametrization for the set of all such interpolants. A more abstract formulation of the analysis leads to a commutant lifting theorem for such multipliers. In particular, we obtain a new proof of a result of Quiggin giving sufficient conditions on a kernel k for a Nevanlinna-Pick-type interpolation theorem to hold for the space of multipliers on a reproducing kernel Hilbert space ℌ (k).