On normal operator logarithms

Abstract

Let X,Y be normal bounded operators on a Hilbert space such that e^X=e^Y. If the spectra of X and Y are contained in the strip S of the complex plane defined by |Im(z)|leq pi, 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 e^X=e^Y. <br />If X is an unbounded self-adjoint operator, which does not have (2k+1) pi, k in Z, as eigenvalues, and Y is normal with spectrum in S satisfying e^{iX}=e^Y, then Y in { e^{iX} }´´. We give alternative proofs and generalizations of results on normal operator exponentials proved by Ch. Schmoeger.