I don't know much about the theory of Hilbert spaces but a research project has me working with them a little bit. In particular requiring an operator to be Hilbert-Schmidt is a recurring condition.

According to wikipedia one nice thing about H-S operators is that on a separable Hilbert space $H$ the set of H-S endomorphisms forms a Hilbert space that is naturally isomorphic to $H\otimes H^*$.

So I'm wondering what else is nice about H-S operators. And what else works with H-S operators that wouldn't work for another class of operators. Fredholm operators also come up a lot. It $T$ is H-S and $S$ is Fredholm what can be said about the composition?