On unitarily equivalent submodules R. DouglasJaydeb Sarkar 46E2246M2047A1347B32Hilbert modulesSilov moduleskernel Hilbert spacesinvariant subspacesisometries The Hardy space on the unit ball in $\mathbb{C}^n$ 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 $\mathbb{C}$ 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. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3406 10.1512/iumj.2008.57.3406 en Indiana Univ. Math. J. 57 (2008) 2729 - 2744 state-of-the-art mathematics http://iumj.org/access/