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.



Does the group of compact perturbations of the identity act transitively on the compact operators?
No. Let $e\_n$ be a basis of the Hilbert space, and let $\alpha\_n$ be any sequence converging to zero. Then you have a compact operator $K\_0$ defined by $K\_0e\_n=\alpha\_ne\_n$. Now let $K\_1$ be any other compact operator. Since compact operators are continuous weak to strong, $K\_1e\_n$ converges to zero. Consequently
$(I+K\_1)K\_0e\_n=\alpha\_n(e\_n+K\_1e\_n)$ converges to zero at the same rate as $\alpha\_n$.


