Let $H$ be a non-separable Hilbert space. Assume $E$ is an orthonormal basis in $H$. Let $E_0=\{e_n\}$ be a countable subset of $E$ and let $\{\zeta_n\}$ be a bounded sequence in $H$. Let $E_1$ be a countable susbet of $E$ and denote $p$ by the projection onto $\overline{span\{e: e\in E_1\}}$.
Question: It seems that, there exists an operator $x\in p^{\perp}B(H)$ satisfying in equations $xe_n=\zeta_n$ for $n=1,2,3,...$, does not it?
An obvious necessary condition is that each $\zeta_n$ is in the span of $E_1$. But, in any case, the operator cannot exist in general: just by constructing a counterexample as in this answer.