compacts subsets of $\mathbb{C}$ can be achieved by compact operators?

Question

If $B \subset \mathbb{C}$ compact. Is it true that given a hilbert space H we can find a bounded operator L on H such that Im(L) = B ? I have a feeling that statement is correct?

Answer 1

Im (L) is a vector space and $B$ is not a vector spaen (except when it is $\{0\}$. So this is impossible.