On normal operator exponentials (Q2759016)

Let \(A\), \(B\) be two bounded normal operators on a complex Hilbert space. Suppose they exponentially commute, i.e. \(e^A e^B= e^B e^A\). The author gives several sufficient spectral conditions under which the operators commute themselves. The principal result is that if both operators satisfy the spectral condition NEWLINE\[NEWLINEE(\sigma(T)\cap \sigma(T+ 2i\pi k))= 0,\quad k= 1,2,\dotsNEWLINE\]NEWLINE (\(E\) -- the spectral measure of \(T\)), then \(AB= BA\). This improves a previous theorem that \(AB= BA\) if both satisfy the stronger condition NEWLINE\[NEWLINE\sigma(T)\cap \sigma(T+ 2i\pi k)= 0,\quad k=1,2,\dotsNEWLINE\]NEWLINE [see the author, Proc. Am. Math. Soc. 127, No. 5, 1337-1338 (1999; Zbl 0914.46037) and also Zbl 0866.39004; Zbl 0964.46029].