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.

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) &mdash; if the <i>n</i>th eigenvalue grows as _n_<sup>_p_</sup> 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 <i>A<sup>-s</sup></i> (=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 &zeta;<sub>_A_</sub>(_s_) = tr(<i>A<sup>-s</sup></i>); 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 &zeta;<sub>_A_</sub>(_s_)) is smooth near _s_=0 and _s_=-1.  All these hopes hold when the eigenvalues of _A_ grow polynomially, whence &zeta;<sub>_A_</sub>(_s_) can be compared to the Riemann zeta function.

Then we can immediately define the "regularized trace" TR _A_ = &zeta;<sub>_A_</sub>(0) and the "regularized determinant" DET _A_ = exp(-&zeta;<sub>_A_</sub>'(0)), where by &zeta;<sub>_A_</sub>'(_s_) I mean the derivative of &zeta;<sub>_A_</sub>(_s_) with respect to _s_.  (If the eigenvalues &lambda;<sub>_n_</sub> are discrete, then &zeta;<sub>_A_</sub>(_s_) = &Sigma; &lambda;<sub>_n_</sub><sup>-_s_</sup>, and so one would have TR _A_ = &Sigma; &lambda;<sub>_n_</sub> and DET _A_ = &Sigma; (log &lambda;<sub>_n_</sub>) &lambda;<sub>_n_</sub><sup>-_s_</sup> |<sub>_s_=0</sub>,  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/d<i>t</i> [ log DET _A_(_t_) ] = TR( A<sup>-1</sup> d<i>A</i>/d<i>t</i> )?  (I can prove this when A<sup>-1</sup>d<i>A</i>/d<i>t</i> 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 d<i>A</i>/d<i>t</i>) are simultaneously diagonalizable (except of course cyclicity), but of course in general they won't commute.