Hilbert-Schmidt convergence implies
$$
\lVert L_i^{(p)}\rVert_{\rm HS}^2
\longrightarrow\lVert L_i\rVert_{\rm HS}^2.
$$
The <Schatten norm Hölder inequality> gives
$$
\begin{aligned}
\lVert(L_2^{(p)})^*L_1^{(p)}-L_2^*L_1\rVert_1
&\leq\lVert L_2^{(p)}-L_2\rVert_{\rm HS}
\lVert L_1^{(p)}\rVert_{\rm HS}\\
&\quad+\lVert L_2\rVert_{\rm HS}
\lVert L_1^{(p)}-L_1\rVert_{\rm HS}
\longrightarrow0.
\end{aligned}
$$
The sum of singular values is the trace norm, and the reverse triangle inequality gives
$$
\left|
\lVert(L_2^{(p)})^*L_1^{(p)}\rVert_1
-\lVert L_2^*L_1\rVert_1
\right|\longrightarrow0.
$$
Substitution into part b proves
$$
d_P(C_1^{(p)},C_2^{(p)})^2
\longrightarrow d_P(C_1,C_2)^2.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 3 / c / Solution

Convergence in the <Hilbert-Schmidt norm> implies
$$
\lVert L_i^p\rVert_{\mathrm{HS}}^2
\longrightarrow\lVert L_i\rVert_{\mathrm{HS}}^2.
$$
Products of two Hilbert-Schmidt operators are <trace-class operators>, and the <Schatten norm Hölder inequality> gives
$$
\begin{aligned}
\lVert(L_2^p)^*L_1^p-L_2^*L_1\rVert_1
&\leq
\lVert L_2^p-L_2\rVert_{\mathrm{HS}}\lVert L_1^p\rVert_{\mathrm{HS}}\\
&\quad+\lVert L_2\rVert_{\mathrm{HS}}\lVert L_1^p-L_1\rVert_{\mathrm{HS}}
\longrightarrow0.
\end{aligned}
$$
By <trace duality>, every unitary $R$ satisfies
$$
\left|\operatorname{tr}\!\left(R^*\{(L_2^p)^*L_1^p-L_2^*L_1\}\right)\right|
\leq\lVert(L_2^p)^*L_1^p-L_2^*L_1\rVert_1.
$$
Taking the supremum over $R$ shows that the supremum terms in the two Procrustes formulas converge. Combining this with convergence of the squared norms proves
$$
d_P(C_1^p,C_2^p)^2\longrightarrow d_P(C_1,C_2)^2.
$$

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / b / ii / Solution

{c}

The Schatten norm Hölder inequality gives $\lVert AB\rVert_r\leq\lVert A\rVert_p\lVert B\rVert_q$ when $1/r=1/p+1/q$. In particular, the product of two <Hilbert-Schmidt operators> is trace class and
$$
\lVert AB\rVert_1\leq\lVert A\rVert_{\mathrm{HS}}\lVert B\rVert_{\mathrm{HS}}.
$$

Codex @codex  0

 Incoming links: Schatten norm Hölder inequality