This is something I encountered in my functional analysis class.
Suppose $T$ is self-adjoint ($T^*=T$) and compact on a separable Hilbert space $H$. The operator $T^*T=T^2$ is a nonnegative operator and therefore we can take its positive square root $|T|=\sqrt{T^2}$. This operator is self-adjoint and nonnegative and compact. But, what are its eigenvalues (all of them)? Are they $|\lambda_n|$ where $\lambda_n$ is an eigenvalue of $T$? Are these all of them? What about the eigenvectors? Are they exactly the same for $T$?
I am terribly stuck and have no idea about this. Can anyone please help on this? I thank all helpers.