Let $H$ be an infinite dimensional (separable if necessary) complex Hilbert space, and denote by $K(H)$ the ideal (in $B(H)$) of compact operators on $H$. Let $G_c=\{I+K\in B(H): I+K \text{ is invertible and } K\in K(H)\}$, and $U_c=\{I+K\in B(H): I+K \text{ is unitary and } K\in K(H)\}$.

$G_c$ and $U_c$ act by left multiplication on $K(H)$. Are these actions transitive?

Additionally, any references which delve into actions of these groups would be appreciated.

EDIT: Clearly, these actions preserve rank, so the answer to the general question is no. But, what I am actually interested in is if they are transitive on each rank-class. In particular, are they transitive on the infinite rank compact operators.