Free outer functions in complete Pick spaces

Abstract

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function f f in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type f = φ g f=\varphi g , where g g is cyclic, φ \varphi is a contractive multiplier, and ‖ f ‖ = ‖ g ‖ \|f\|=\|g\| . In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.