  The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators
combined with the Beurling-Lax theorem establishes a correspondence between any
two of four kinds of objects: shift-invariant subspaces, operator-valued inner
functions, conservative discrete-time input/state/output linear systems, and
$C_{\cdot 0}$ Hilbert-space contraction operators. We discuss an analogue of
all these ideas in the context of weighted Hardy spaces over the unit disk and
an associated class of hypercontraction operators.
