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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.

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  • $\begingroup$ In finite dimensions, the first representation is a singular value decomposition, while the second is a matrix representation with respect to some orthonormal basis. These are different things. $\endgroup$ – Daniel Fischer Sep 7 '15 at 10:00
  • $\begingroup$ @danielfischer: Is the matrix representation correct? $\endgroup$ – niki Sep 7 '15 at 10:19
  • $\begingroup$ Not sure. Is $\xi \otimes \eta$ the map $w \mapsto \langle w,\xi\rangle\cdot \eta$ or $w \mapsto \langle w,\eta\rangle\cdot \xi$? And there may be convergence issues, I'm not sure in which topologies the double series converges (I'm almost sure it converges in the weak operator topology, and most likely also in the strong operator topology, I'm uncertain about the norm topology). But basically, it looks alright. $\endgroup$ – Daniel Fischer Sep 7 '15 at 11:14
  • $\begingroup$ @danielfischer : second one is correct $\endgroup$ – niki Sep 7 '15 at 11:38
  • $\begingroup$ Then I think you must swap some indices. As written, we'd have (assuming everything converges nicely) $$x(e_k) = \sum_{i,j} \langle xe_j,e_i\rangle (e_j \otimes e_i)(e_k) = \sum_j \langle xe_j,e_k\rangle e_j,$$ which generally isn't correct. $\endgroup$ – Daniel Fischer Sep 7 '15 at 11:46

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