Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 106 / 3 / b / iv / Solution

Let $T=AB$ with $A,B$ Hilbert--Schmidt, and write $T=U|T|$. Since $|T|=U^{\ast }AB$, part iii and the Cauchy-Schwarz inequality for series giveThus $T$ is trace class with the required bound.