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.