Dilations, Wandering Subspaces, and Inner Functions

The objective of this paper is to study wandering subspaces for commuting tuples of bounded operators on Hilbert spaces. It is shown that, for a large class of analytic functional Hilbert spaces ${\mathcal{H}}_K$ on the unit ball in ${\mathbb{C}}^n$, wandering subspaces for restrictions of the multiplication tuple $M_z=(M_{z_1},\dots ,M_{z_n})$ can be described in terms of suitable ${\mathcal{H}}_K$-inner functions. We prove that ${\mathcal{H}}_K$-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogenous polynomials as an application. Along the way we prove a refinement of a result of Arveson on the uniqueness of minimal dilations of pure row contractions.