This is related to [Theo's question](https://mathoverflow.net/questions/2677/abelianization-of-lie-groups) about the abelianizations of finite dimensionsal Lie groups.


I am interested in a specific (infinite-dimensional) case of the above question. Let H be an infinite-dimensional Hilbert space and GL(H) represent the bounded two-sided invertible operators on H. Is there a nice description of the commutator subgroup G (the group generated be elements of the form ABA^{-1}B^{-1}) and the abelianization GL(H)/G? 



Abelianization of GL(H)
The solution to Problem 240 in Halmos is: "On an infinite-dimensional Hilbert space, the commutator subgroup of the full linear group is the full linear group itself."


It is mentioned, although the details are not given, that every invertible operator is the product of two(!) commutators. Reference: A. Brown and C. Pearcy, *Multiplicative commutators of operators*, Can. J. Math. **18** (1966) 737-749.


