There is a nice formula, called the Plemelj-Smithies formula that allows the computation of the determinant of certain operators in terms of the trace. I'm including this as an answer because it holds also in infinite dimensions.

See, for example, section 1.2 of the paper [here][1]. This is a paper by Gohberg, Goldberg and  Krupnik that eventually they extended into a very nice book.  


  [1]: http://krein.unica.it/~cornelis/private/PDF/IEOT/ieot_26_136.pdf