Nevanlinna-Pick Interpolation and Factorization of Linear Functionals

TL;DR

This paper establishes a Nevanlinna-Pick interpolation framework for certain operator algebras on reproducing kernel Hilbert spaces, including Bergman and Hardy spaces, using properties of their multiplier algebras.

Contribution

It introduces a Nevanlinna-Pick family of kernels for unital weak-* closed algebras with property A_1(1), extending interpolation results to a broad class of function spaces.

Findings

Nevanlinna-Pick interpolation applies to many function spaces over the unit disk.

Multiplier algebra of a complete NP space has property A_1(1).

Matrix version of the interpolation theorem is established.

Abstract

If $\fA$ is a unital weak-$*$ closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property $\bA_1(1)$, then the cyclic invariant subspaces index a Nevanlinna-Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has $\bA_1(1)$, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-$*$ closed subalgebras of $H^\infty$ acting on Hardy space or on Bergman space.