Abstract. In a previous paper the author proved that for square matrices with algebraic entries exp(A)exp(B) =exp(B)exp(A) if and only if AB = BA. This result is extended here to bounded operators on an arbitrary Banach space.The simple and well-known fact that To prove the 'only if' part of (1), observe that if A is selfadjoint, then e A is positive, and the positive square root e A/2 can be approximated by a sequence of polynomials in e A (for a simple iterative construction of suitable polynomials, due to C. Visser, see, e.g., [8, p. 261f.], or [16, p. 222ff.]; a more general statement is [4, Theorem 11.3.5] n for all n ∈ N. Expanding the power series on both sides leads towhence AB = BA. The second equivalence is a trivial consequence of the first one, since (e A e B ) * = e B eA .
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