Fact : I know for every compact operator $x : H \to H$, there are sequences $\{\xi_n\}$ and $\{\eta_n\}$ of orthonormal vectors of separable Hilbert space $H$, and sequence $\{\alpha_n\}$ in $\Bbb C$ such that $$x = \sum_{n=1}^\infty \alpha_n \xi_n\otimes \eta_n$$ Let $\{e_n\}$ be an orthonormal basis for $H$.
I think matrix $(\langle xe_j,e_i\rangle)_{i,j\in \Bbb N}$ is a matrix representation of compact operator $x$, and $$x = \sum_{i,j=1}^\infty \langle xe_j,e_i\rangle e_i\otimes e_j$$ is a representation of $x$. In this case, for every orthonormal system, we can write a representation of $x$. But by above fact, for a representation of $x$, there are orthonormal vectors. Where is my mistake?
Thanks in advance.