Every compact operator is a commutator of compact operators

We prove that every compact operator $T$ on a separable infinite-dimensional complex Hilbert space is a commutator of two compact operators, thereby answering a question of Pearcy and Topping. Moreover, the compact factors $A$ and $B$ can be chosen such that $[A,B]=T$ and $\max \{\mathrm{\parallel }A\mathrm{\parallel },\mathrm{\parallel }B\mathrm{\parallel }\}\le c\mathrm{\parallel }T\mathrm{\parallel }{}^{1\mathrm{/}2}$ for a universal constant $c$.