Compact and self-adjoint operator It is true that if $T:H \to H$ is a compact operator ($H$ Hilbert space) then $T^\ast T$ is algo compact and indeed self-adjoint.
Conversely, is it true that every compact and self-adjoint operator $S$ can be decomposed like $S=A^\ast A$ with $A$ compact?
Thanks!

No, an operator of the form $A^*A$ is selfadjoint and has non-negative spectrum. So any compact selfadjoint operator with a negative eigenvalue cannot be written in that form. 
For the simplest example, fix a vector $x\in H$ and let $S$ be the operator
$$
Sy=-\langle y,x\rangle x.
$$
