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.

As Mike notes Hilbert bundles are trivial over most spaces.

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

Studies the homotopy type of the relative Grasmannian 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.