In this talk we will discuss an analytic model theory for pure hyper-
contractions (introduced by J. Agler) which is analogous to Sz.Nagy-Foias
model theory for contractions. We then proceed to study analytic model
theory for doubly commuting n-tuples of operators and analyze the
structure of joint shift co-invariant subspaces of reproducing kernel
Hilbert spaces over polydisc. In particular, we completely characterize
the doubly commuting quotient modules of a large class of reproducing
kernel Hilbert Modules, in the sense of Arazy and Englis, over the unit
polydisc.
Inspired by Halmos, in the second half of the talk, we will focus on the
wandering subspace property of commuting tuples of bounded operators on
Hilbert spaces. We prove that for a large class of analytic functional
Hilbert spaces H_k on the unit ball in C^n, wandering subspaces for
restrictions of the multiplication tuple M_z = (M_{z_1},...,M_{z_n}) can
be described in terms of suitable H_k-inner functions. We also prove
that H_k-inner functions are contractive multipliers and deduce a result
on the multiplier norm of quasi-homogeneous polynomials as an
application. Along the way we also prove a refinement of a result of
Arveson on the uniqueness of the minimal dilations of pure row
contractions.