Thompson-type formulae

Let X and Y   be two n×nn×n Hermitian matrices. In the article Proof of a conjectured exponential formula   (Linear Multilinear Algebra 19 (1986) 187–197) R.C. Thompson proved that there exist two n×nn×n unitary matrices U and V such thateiXeiY=eiUXU⁎+VYV⁎.eiXeiY=eiUXU⁎+VYV⁎. In this note we consider extensions of this result to compact operators as well as to operators in an embeddable II1 factor.