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?



Hilbert Schmidt operators
I think the perfect reference for you is Lars Hörmander's *The Analysis of Linear Partial Differential Operators*, vol III (the chapter on elliptic operators). There you'll find in perfect Hörmander style all you need about Fredholm, Hilbert-Schmidt, trace class operators. (As to the composition, H-S is a two-sided ideal; if S is any bounded operator TS and ST are H-S). 


