This is a Banach space version of Andre Henriques question
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ are both nuclear. Assume whatever approximation properties on $X$ and $Y$ that you want (say, assume that both $X^*$ and $Y^*$ have the bounded or even metric approximation property), so that the trace of $ab$ and of $ba$ are well defined. Then must $tr(ab)=tr(ba)$?
When $X$ and $Y$ are Hilbert spaces, you can find three correct proofs and one interesting but incomplete proof at the above link. None of these generalize immediately to the Banach space setting.
Caveat: I have not done a literature search or thought much about this problem, but it is natural to consider it after reading Andre's question.