Let $H$ be a non-separable Hilbert space and $E$ be an orthonormal basis in $H$. Let $E_0$ be a countable subset of $E$ and $p$ be the projection onto the space generated by $E_0$.
Let $\{\zeta_n\}$ be an arbitrary sequence in $H$ and $\{r_n\}$ a sequence of strictly positive numbers.
Question: For given a bounded sequence $\{\eta_n\}$ in $H$, I am looking for an operator $x\in p^{\perp}B(H)$ satisfying in $||x\eta_n-\zeta_n||=r_n$.