  The Hardy space on the unit ball in C^n provides examples of a quasi-free,
finite rank Hilbert module which contains a pure submodule isometrically
isomorphic to the module itself. For n=1 the submodule has finite codimension.
In this note we show that this phenomenon can only occur for modules over
domains in the complex plain and for finitely-connected domains only for
Hardy-like spaces, the bundle shifts. Moreover, we show for essentially
reductive modules that even when the codimension is infinite, the module is
subnormal and again, on nice domains such as the unit ball, must be Hardy-like.
