  For $k=1,\ldots,K$, let $A_k$ and $B_k$ be positive semidefinite matrices
such that, for each $k$, $A_k$ commutes with $B_k$. We show that, for any
unitarily invariant norm, \[ |||\sum_{k=1}^K A_kB_k||| \le ||| (\sum_{k=1}^K
A_k)\;(\sum_{k=1}^K B_k)|||. \]
