When can we exchange the trace and an integral/limit/derivative? For a trace class operator $A$ (acting on a Hilbert space), that is parameterised by a real variable $x$, what are the conditions for the following to hold?
$$ \mathrm{tr} \int_a^b A(x) \, dx = \int_a^b \mathrm{tr} \, A(x) \, dx $$
$$ \mathrm{tr} \lim_{x \to a} A(x) = \lim_{x \to a} \mathrm{tr} \, A(x) $$
$$ \mathrm{tr} \frac{d A(x)}{dx} = \frac{d}{dx} \mathrm{tr} \, A(x) $$
I guess you could always pick a basis, write the expressions as infinite sums and use uniform convergence as a criterium. Is there a more economical way?

The trace is a continuous linear functional on the space of trace-class operators.
So you can do what you want provided that $\int_a^b A(x)dx$, $\lim_{x\to a} A(x)$ and $\frac{dA(x)}{dx}$ make sense with respect to the trace-class norm. For the integral, this means that $x\mapsto A(x)$ should be for example Bochner-integrable from $[a,b]$ into the space of trace-class operators; and for the other two things, that the limits should exist with respect to the trace-class norm.
Presumably, there are also weaker conditions that make everything work.
