Let $H$ be separable Hilbert space.I’m trying to prove that
The only proper closed ideal of $B(H)$ is $K(H)$, the compact operators on $H$
I can see that every closed ideal of $B(H)$ must contain $K(H)$ but I cannot see the other part.
I’m sure that that there must be many books which contains proof of this result but I could not find any. Any references which has proper proof of the same fact?