Nevanlinna-pick Interpolation and Factorization of Linear Functionals

If A is a unital weak-∗ closed subalgebra of a multiplier algebra on a reproducing kernel Hilbert space which has property A1(1), then the cyclic invariant subspaces provide an Nevanlinna-Pick (NP) family of kernels. This yields an NP interpolation theorem for a wide class of operator algebras. In particular, it applies to many kernel spaces over the unit disk including the Bergman space. We also show that the multiplier algebra of a complete NP space has A1(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∞ acting on Hardy space or on Bergman space.