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.

Let A be an operator (on an infinite-dimensional vector space). You might as well assume that its spectrum is all real and positive. In fact, I only care when the spectrum is discrete and grows polynomially, but I hear that this stuff works more generally.

In general, A is not trace-class (the sum of the eigenvalues converges) or determinant-class (the product of the eigenvalues converges) — if the nth eigenvalue grows as n^p for some p>0, then it won't be. But there is a procedure to try to define a "trace" and "determinant" of A nevertheless.

Let us hope that for large enough s, the operators A^-s (=exp(-s log A), and log A makes sense if the spectrum of A is positive) are trace-class. If so, then we can define ζ_A(s) = tr(A^-s); it is analytic for Re(s) large enough. Let's hope that it has a single-valued meromorphic continuation and that this function (which I will also call ζ_A(s)) is smooth near s=0 and s=-1. All these hopes hold when the eigenvalues of A grow polynomially, whence ζ_A(s) can be compared to the Riemann zeta function.

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).

So, here are my questions:

1. Is it true that exp TR A = DET exp A?
2. Let A(t) be a smooth family of operators (t is a real variable). Is it true that d/dt [ log DET A(t) ] = TR( A^-1 dA/dt )? (I can prove this when A^-1dA/dt is trace-class.)
3. Is DET multiplicative, so that DET(AB) = DET A DET B? (I can prove this using 1. and 2., or using the part of 2. that I can prove if B is determinant-class.)
4. Is TR cyclic, i.e. TR(AB) = TR(BA)?
5. Is TR linear, i.e. TR(A + B) = TR A + TR B?

None of these are even obvious to me when A and B (or dA/dt) are simultaneously diagonalizable (except of course cyclicity), but of course in general they won't commute.