Let $H$ be an infinite dimensional separable complex Hilbert space with Lie group action (I am mostly interested in the case $S^1$). Let $\text{Gl}_{G}(H)$ be the space of invertible, bounded and equivariant linear maps (from $H$ to $H$).

Now, in the non-equivariant case, Kuiper's theorem states that $\text{Gl}(H)$ is (weakly) contractible. Is this this also true for $\text{Gl}_{G}(H)$?