Optimal commuting approximation of Hermitian operators

We formulate and provide a solution to an approximation problem that occurs in various settings: Finding an optimal additive decomposition of a given Hermitian Hilbert-Schmidt operator, in a term commuting with a second Hermitian compact operator and a term as small as possible in the trace norm sense. In the finite-dimensional case, we show how to interpret our result through a Sylvester equation. An application to a quantum information problem and an interpretation in quantum probability are also sketched.