It is well known that $K(H)$, the set of compact operators on a separable Hilbert space $H$ is the only two-sided closed ideal in $B(H)$, the algebra of all bounded linear operators.
Ar there any closed one-sided ideal in $B(H)$ other than $K(H)$? In fact, can anyone give me an example of a one-sided but not two-sided ideal in $B(H)$, not necessarily closed?
If $K\subset H$ is a subspace, then $\{T\in B(H)\mid T(H)\subset K\}$ is a right-sided ideal. It is closed if $K$ is a closed subspace. Similarly you get left-sided ideals by considering all operators whose kernel contains a given subspace. Except for the trivial cases, these are no two-sided ideals.