We prove that K₁ of the compact operators is zero. This theorem has the following operator-theoretic formulation: any invertible operator of the form (identity) + (compact) is the product of (at most eight) multiplicative commutators (AjBjAj⁻¹Bj⁻¹)±1, where each Bj is of the form (identity) + (compact). The proof uses results of L. G. Brown, R. G. Douglas, and P. A. Fillmore on essentially normal operators and a theorem of A. Brown and C. Pearcy on multiplicative commutators.
L.G. Brown and C. Schochet, K₁ of the compact operators is zero, Proceedings of the American Mathematical Society 59(1) (1976), 119-122.