= Unitary polar factor of a finite compression
Let $P_n:H\to H_n$ be coisometries onto finite-dimensional <Hilbert spaces>, with $Q_n=P_n^*P_n\to I$ strongly, and let $A$ be a <unitary operator> on $H$. The square compression $T_n=P_nAP_n^*$ has a <singular value decomposition> $T_n=V_n\Sigma_nW_n^*$. Thus $U_n=V_nW_n^*$ 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 $H$ is
$$
P_n^*U_nP_n\longrightarrow A\quad\text{strongly}.
$$
Indeed, $P_n^*T_n^*T_nP_n=Q_nA^*Q_nAQ_n\to I$ strongly, with uniformly bounded <operator norms>. Polynomial approximation to the square-root function gives $P_n^*|T_n|P_n\to I$ strongly. Since $T_n=U_n|T_n|$,
$$
\|(U_n-T_n)P_nx\|=\|(I-|T_n|)P_nx\|\to0.
$$
Finally, $P_n^*T_nP_n=Q_nAQ_n\to A$ strongly. Including the lift $P_n^*$ makes these comparisons between operators on the same <Hilbert space>.
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