Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$.

Let $E$ be a norm closed convex subset of positive operators in $K(H)$ and let $a$ be a non-zero positive compact operator where $a\notin E$.

Q: Is there any vector $\zeta\in H$ which separates $a$ and $E$, I mean there is a a positive number $\lambda$ such that for all $x\in E$ $$ \langle x\zeta,\zeta\rangle\leq \lambda< \langle a\zeta,\zeta\rangle$$