Unitary polar factor of a finite compression

Let $P_n:H\to H_n$ be coisometries onto finite-dimensional Hilbert spaces, with $Q_n=P_n^{\ast }{P}_n\to I$ strongly, and let $A$ be a unitary operator on $H$. The square compression $T_n=P_{n}A{P}_n^{\ast }$ has a singular value decomposition $T_n=V_n{\Sigma }_n{W}_n^{\ast }$. Thus $U_n=V_n{W}_n^{\ast }$ is a unitary operator on $H_n$ extending the polar factor, including a choice on its null space.

Because the codomains vary, the convergence statement in $H$ isIndeed, $P_n^{\ast }{T}_n^{\ast }{T}_n{P}_n=Q_n{A}^{\ast }{Q}_{n}A{Q}_n\to I$ strongly, with uniformly bounded operator norms. Polynomial approximation to the square-root function gives $P_n^{\ast }|T_n|P_n\to I$ strongly. Since $T_n=U_n|T_n|$,Finally, $P_n^{\ast }{T}_n{P}_n=Q_{n}A{Q}_n\to A$ strongly. Including the lift $P_n^{\ast }$ makes these comparisons between operators on the same Hilbert space.