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 $x\in E$
$$ \langle x\zeta,\zeta\rangle\leq \lambda< \langle a\zeta,\zeta\rangle$$



A point-wise separation Hahn-Banach theorem in C*-algebras
$2 \times 2$ counterexample, $E = \left\{\left[\matrix{\lambda& 0\cr 0&\lambda}\right]: \lambda \geq 0\right\}$ and $A = \left[\matrix{1&1\cr 1&1}\right]$. Then for any nonzero $\zeta$ we have $\{\langle B\zeta,\zeta\rangle: B \in E\} = [0,\infty)$, so no $\zeta$ can separate.


The general idea is that you can separate $E$ and $a$ with a bounded linear functional on $K(H)$, i.e., tracing against some trace class operator, but you can't expect to do it with a single vector.


