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 HK on the unit ball in Cn, wandering subspaces for restrictions of the multiplication tuple Mz=(Mz1,…,Mzn) can be described in terms of suitable HK-inner functions. We prove that HK-inner functions are contractive multipliers and deduce a result on the multiplier norm of quasi-homogeneous 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.