The paper
MR2481802 (2010e:46083) Reviewed Abbondandolo, Alberto (I-PISA); Majer, Pietro (I-PISA) Infinite dimensional Grassmannians. (English summary) J. Operator Theory 61 (2009), no. 1, 19–62. 46T05 (47A53 58B15)
https://arxiv.org/pdf/math/0307192.pdf
Contains similar results when the projector is compact. It has been too long since I've thought about Hilbert Schmidt operators to see if this is the same answer.