Let $H$ be a Hilbert space and $T:H \rightarrow H$ a compact operator, show that $$T=\sum_{n=1}^{\infty} \langle x,e_n \rangle v_n$$ where $\{e_n\}$ is a orthonormal basis of $H$ and $\{v_n\}$ is a orthogonal set.
I try to emulate the prove of the spectral descomposition theorem but i dont sure to these is the right way, the hint of the book is "see the operator $TT^{*]$ but i dont see how ocupate these fact. Any hint or suggestion i will very grateful.