Nevanlinna-Pick Interpolation and Factorization of Linear Functionals

If $\text{fA}$ is a unital weak-$\ast$ closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property ${\text{bA}}_1(1)$, 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 ${\text{bA}}_1(1)$, 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-$\ast$ closed subalgebras of $H^{\mathrm{\infty }}$ acting on Hardy space or on Bergman space.