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On commuting exponentials in low dimensions. (English) Zbl 1119.15023

We deal with square matrices \(A, B\) of dimension \(d = 2\) or \(3\), over the complex field, such that \(AB\neq BA\). We introduce the relations \(G_t:\exp(tA+B)=\exp(tA)\exp(B)\) and \(G_t':\exp(tA+B)= \exp(tA)\exp(B)=\exp(B)\exp(tA)\). In dimension 2, we characterize the \((A, B)\) couples satisfying \(G_t\) for any \(t \in \mathbb N\). In dimension 2 or 3, we show that if \(G_t'\) is satisfied for any \(t \in \mathbb N\) then \(A\) and \(B\) are simultaneously triangularizable. In this manner we do not need the \(2i\pi\)-congruence-free postulate anymore, which has been supposed by researchers since 1954.

MSC:

15A54 Matrices over function rings in one or more variables
39B42 Matrix and operator functional equations
15A27 Commutativity of matrices
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References:

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