Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 358 / 2 / d / Solution

Let $P_n:{\ell }^2(\mathbb{N})\to {\mathbb{C}}^n$ be coordinate projection and setThis finite matrix is computable from the matrix entries of $A$. Compute its singular value decompositionand define the unitary polar factor of a finite compressionThis is a unitary operator on ${\mathbb{C}}^n$, including when $T_n$ is singular.

Put $Q_n=P_n^{\ast }{P}_n$. Since $Q_n\to I$ strongly and $A$ is unitary,The continuous functional calculus for positive matrices therefore givesThe polar identity $T_n=A_n|T_n|$ now yieldsAlso $P_n^{\ast }{T}_n{P}_n=Q_{n}A{Q}_n\to A$ strongly, and henceIn particular the weak convergence required in part (c) holds. The construction uses only a finite block of the given matrix and a finite singular value decomposition, so it is an algorithm realizing all the assumptions of part (c).