Let $(X_N)_N$ be a sequence of trace class operators acting on, say, $L^2(\mathbb{R})$. What are the minimal assumptions in order to have the convergence of their Fredholm determinant $$ \lim_N\det(I+X_N) ? $$ I know $X\mapsto \det(I+X_N)$ is continuous for the trace class norm topology (once restricted to trace class), ok. Let's say that $X_N$ converges weakly to $X$, it is enough to have the convergence of Fredholm determinants ? What conditions should we add ? I guess I just need a good reference on the topic. Any ideas ? Thanks in advance.
The Fredholm determinant is not sequentially continuous in the strong topology. Take the sequence of 1dimensional projectors $X_n=\langle e_n,\cdot\rangle e_n$ with the $e_n$ forming an ON basis. You then have $X_n \to 0$ strongly, but $0=\det(IX_n)$ does not converge to $1=\det(I)$. Or what did you have in mind when saying that you would know the continuity for the strong topology (once restricted to trace class)?
If you, however, add the convergence $\X_n\_1\to\X\_1$, where $\\cdot\_1$ denotes trace class norm, then it is known that weak sequential convergence $X_n \to X$ implies convergence in trace class norm and, therefore, of the Fredholm determinant (see Thm. 2.21 and Addendum H of the book "Trace Ideals and Their Applications" by Barry Simon, 2nd ed., AMS 2005).

$\begingroup$ Sorry, I meant "trace class norm"! I'm editing my post ... Thx ! $\endgroup$ – Adrien Hardy Aug 12 '11 at 10:32

$\begingroup$ Folkmar, do you know other results if one restrict to operators which are symmetric positive integral kernel operators ? Anyway, thx for your answer $\endgroup$ – Adrien Hardy Aug 12 '11 at 12:07

$\begingroup$ Weak operator convergence and convergence of the trace (which is then, in the case of symmetric positive integral trace class operators, just the trace class norm) will do. Have a look at Chapter 2 of Simon's book. $\endgroup$ – Folkmar Bornemann Aug 12 '11 at 15:59

1$\begingroup$ Does convergence of the trace also suffice in the case that the operators in question are not positive symmetric? This statement is not contained in the book of Simon, I believe! $\endgroup$ – Matthias Ludewig Jan 20 '16 at 13:53