On the operator equation eA=eB

Suppose that A and B are bounded linear operators on a complex Hilbert space and that eA=eB. It is well-known that if the spectrum of A is incongruent (mod 2πi) then AB=BA. In this note we show that if A is normal and A π then eA=eB implies that A2B=BA2. If B is also normal, B π and −iπ is not an eigenvalue of A then we show that eA=eB implies AB=BA and (A−B)2=2πi(A−B).