Can one give an example of two bounded operators A and B in a Hilbert space such that both products AB and BA are of the trace class but their traces are different? If one of them is compact then the traces are equal; if one requires the commutator [A,B] to be of the trace class, not each product separately then the left shift and the right shift give an example.



trace of a commutator
See


[tr(ab) = tr(ba)?](https://mathoverflow.net/questions/76386/trab-trba)


and 


[tr(ab)=tr(ba), part 2.](https://mathoverflow.net/questions/76604/trab-trba-part-2) 


for the Banach space version.


