On commuting operator exponentials

Let $A$, $B$ be bounded operators on a Banach space with $2\pi i$-congruence-free spectra such that $e^Ae^B=e^Be^A$. E. M. E. Wermuth has shown that $AB=BA$. Ch. Schmoeger later established this result, using inner derivations and, in a second paper, has shown that: for $a,b$ in a complex unital Banach algebra, if the spectrum of $a+b$ is $2\pi i$-congruence-free and $e^ae^b=e^{a+b}=e^be^a$, then $ab=ba$ (and thus, answering an open problem raised by E. M. E. Wermuth). In this paper we use the holomorphic functional calculus to give alternative simple proofs of both of these results. Moreover, we use the Borel functional calculus to give new proofs of recent results of Ch. Schmoeger concerning normal operator exponentials on a complex Hilbert space, under a weaker hypothesis on the spectra.