Corollary 3

Let v be a continuous positive endomorphism of the hilbertian space E with a finite trace. There exists a positive Hilbert-Schmidt endomorphism w of E such that $v = w^2$ and such that v commutes with w.

With the notations of cor. 1, it is enough to consider the endomorphism w which transforms the vector $\sum_{i \in I} \xi_i e_i$ into the vector $\sum_i \lambda_i^{1/2} \xi_i e_i$.