Nevanlinna-Pick Interpolation and Factorization of Linear Functionals

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.Comment: 26 pages; minor revisions; to appear in Integral Equations and Operator Theor