Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis for $H$. Let $E_0$ be a countable subset of $E$ and $\{\delta_i\}_1^{\infty}$ be a bounded sequence of $(0,\infty)$. For given an arbitrary sequence $\{h_i\}_1^{\infty}$ of unit vectors in $H$, I am looking for an operator $y$ in $B(H)$ satisfying in the following properties:
1- $\langle ye,f\rangle=0$ for all $e,f$ in $E_0$.
2- $||yh_i||\geq\delta_i$ for every $i\geq1$
I feel the following argument works:
Let $E_0'$ be a countable subset of $E$ containing $E_0$ such that $h_i's$ are all in the closure of span$E_0'$. Let $F$ be a countable subset of $E$ with $F\cap E_0'$ is empty. If $\delta=\sup \delta_i$, then the following operator works well $$y=M\sum_{f\in F,e\in E_0'}f\otimes e$$