Schröder-Bernstein theorem for Hilbert space modules
= Schröder-Bernstein theorem for Hilbert space modules
{c}
Two <Hilbert space modules> are unitarily equivalent if each embeds isometrically as a closed submodule of the other. For isometric intertwiners $u:H_1\to H_2$ and $v:H_2\to H_1$, let $D=H_1\ominus vH_2$ and $K=\bigoplus_{n\geq0}(vu)^nD$. These summands are orthogonal. The operator acting as $u$ on $K$ and as $v^*$ on $K^\perp$ is the required unitary onto $H_2$.