On normal operator logarithms

Let X,Y be normal bounded operators on a Hilbert space such that eX=eY. If the spectra of X and Y are contained in the strip S of the complex plane defined by |I(z)|⩽π, we show that |X|=|Y|. If Y is only assumed to be bounded, then |X|Y=Y|X|. We give a formula for X-Y in terms of spectral projections of X and Y provided that X,Y are normal and eX=eY. If X is an unbounded self-adjoint operator, which does not have (2k+1)π,k∈Z, as eigenvalues, and Y is normal with spectrum in S satisfying eiX=eY, then Y∈{eiX}″. We give alternative proofs and generalizations of results on normal operator exponentials proved by Schmoeger.