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There is a nice formula, called the Plemelj-Smithies formula that allows the computation of the determinant of certain operators in terms of the tracemoments (Tr(A^n)) of A. I'm including this as an answer because it the formula organizes the information nicely for all finite dimensions, and holds also in infinite dimensions.

See, for example

For the formula, see section 1.2 of the paper here. This is a paper by Gohberg, Goldberg and Krupnik that eventually they extended into a very nice book.

Later on, I'll try to come back and type in the formula...

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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. This is a paper by Gohberg, Goldberg and Krupnik that eventually they extended into a very nice book.

Regarding going from determinants back to traces, you'll have to look in the book.

1

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. This is a paper by Gohberg, Goldberg and Krupnik that eventually they extended into a very nice book.

Regarding going from determinants back to traces, you'll have to look in the book.