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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?

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up vote 3 down vote accepted

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).

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Oh Pietro, would it really hurt to type out "Analysis of Linear Partial Differential Operators"? =) Another good reference is Reed and Simon, "Methods of Mathematical Physics" vol 1 Chapter VI. –  Willie Wong Aug 4 '10 at 1:49
    
sorry, fixed now –  Pietro Majer Aug 4 '10 at 5:48
    
Generalized functions 4 of Gelfand and Shilov is good as well –  user36539 Sep 14 '13 at 23:38
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