Every compact operator is a commutator of compact operators

Abstract

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 $\max\{\Vert{}A\Vert{},\Vert{}B\Vert{}\}\leq c\Vert{}T\Vert{}^{1/2}$ for a universal constant $c$.