Which operators are the self-commutators of compact operators?

Abstract:

It is shown that, for a hermitian operator T on a Hubert space H, the following three statements are equivalent: (i) $T = {A^ \ast }A - A{A^ \ast }$ for some compact operator A, (ii) there is an orthonormal basis $\{ {b_j}\}$ such that $\langle T{b_j},{b_j}\rangle = 0$ for all j, and (iii) ${\text {tr}} {T^ + } = {\text {tr}} {T^ - }$ (possibly infinite) where ${T^ + } = (|T| + T)/2$ and ${T^ - } = (|T| - T)/2$.