Unitary polar factor of a finite compression

If $T_n=P_{n}A{P}_n^{\ast }$ is a square finite compression with singular value decomposition $T_n=V_n{\Sigma }_n{W}_n^{\ast }$, then $U_n=V_n{W}_n^{\ast }$ is a unitary extension of its polar factor. When $A$ is unitary and $P_n^{\ast }{P}_n\to I$ strongly, $|T_n|\to I$ strongly and hence $U_n{P}_n\to A$ strongly.