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Then we can immediately define the "regularized trace" TR A = ζ_A(-1) and the "regularized determinant" DET A = exp(-ζ_A'(-1)), where by ζ_A'(s) I mean the derivative of ζ_A(s) with respect to s. (If the eigenvalues λ_n are discrete, then ζ_A(s) = Σ λ_n^-s, and so one would have TR A = Σ λ_n and DET A = Σ (log λ_n) λ_n^-s |_s=0, if they converged.) If A is trace- (determinant-) class, then TR A = tr A (DET A = det A).

Then we can immediately define the "regularized trace" TR A = ζ_A(0) and the "regularized determinant" DET A = exp(-ζ_A'(0)), where by ζ_A'(s) I mean the derivative of ζ_A(s) with respect to s. (If the eigenvalues λ_n are discrete, then ζ_A(s) = Σ λ_n^-s, and so one would have TR A = Σ λ_n and DET A = Σ (log λ_n) λ_n^-s |_s=0, if they converged.) If A is trace- (determinant-) class, then TR A = tr A (DET A = det A).

The short answer to may question my be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.

The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.