I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
If you are interested in closed ideals, then
If you are inrested in all ideals, then they are between $F(H)$ and $K(H)$ and consisist of operators whose singular values belong to some order ideal in $c_0^+$.
For details see section I.8.7 here.