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

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