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

Table of contents
- Solution ii

Solution

Convergence in the Hilbert-Schmidt norm implies

∥ L_{i}^{p} ∥_{HS}^{2} ⟶ ∥ L_{i} ∥_{HS}^{2} .

Products of two Hilbert-Schmidt operators are trace-class operators, and the Schatten norm Hölder inequality gives

∥(L_{2}^{p})^{*} L_{1}^{p} - L_{2}^{*} L_{1} ∥_{1} \leq ∥ L_{2}^{p} - L_{2} ∥_{HS} ∥ L_{1}^{p} ∥_{HS} + ∥ L_{2} ∥_{HS} ∥ L_{1}^{p} - L_{1} ∥_{HS} ⟶ 0.

By trace duality, every unitary

R

satisfies

∣ tr (R^{*} {(L_{2}^{p})^{*} L_{1}^{p} - L_{2}^{*} L_{1}}) ∣ \leq ∥(L_{2}^{p})^{*} L_{1}^{p} - L_{2}^{*} L_{1} ∥_{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} ⟶ d_{P} (C_{1}, C_{2})^{2} .

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