Bourgeois, Gerald On commuting exponentials in low dimensions. (English) Zbl 1119.15023 Linear Algebra Appl. 423, No. 2-3, 277-286 (2007). 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. Reviewer: Grosio Stanilov (Sofia) Cited in 4 Documents MSC: 15A54 Matrices over function rings in one or more variables 39B42 Matrix and operator functional equations 15A27 Commutativity of matrices Keywords:commuting exponentials; congruence-free spectrum; simultaneously triangularizable PDFBibTeX XMLCite \textit{G. Bourgeois}, Linear Algebra Appl. 423, No. 2--3, 277--286 (2007; Zbl 1119.15023) Full Text: DOI arXiv References: [1] Baribeau, L.; Roy, S., Caractérisation spectrale de la forme de Jordan, Linear Algebra Appl., 320, 183-191 (2000) · Zbl 0972.15003 [2] Hille, E., On roots and logarithms of elements of a complex Banach algebra, Amer. Math. Ann., 136, 46-57 (1958) · Zbl 0081.11203 [3] Morinaga, K.; Nono, T., On the non-commutative solutions of the exponential equation \(e^x e^y = e^{x + y}\), J. Sci. Hiroshima Univ. (A), 17, 345-358 (1954) · Zbl 0056.01506 [4] Morinaga, K.; Nono, T., On the non-commutative solutions of the exponential equation \(e^x e^y = e^{x + y}\), II, J. Sci. Hiroshima Univ. (A), 18, 137-178 (1954) · Zbl 0057.25103 [5] Paliogiannis, F. C., On commuting operator exponentials, Proc. Amer. Math. Soc., 131, 3777-3781 (2003) · Zbl 1069.47018 [6] Schmoeger, Ch., Remarks on commuting exponentials in Banach algebras, Proc. Amer. Math. Soc., 127, 5, 1337-1338 (1999), MR 99h:46090 · Zbl 0914.46037 [7] Schmoeger, Ch., Remarks on commuting exponentials in Banach algebras II, Proc. Amer. Math. Soc., 128, 11, 3405-3409 (2000), MR 2001b:46077 · Zbl 0964.46029 [8] Wermuth, E. M.E., Two remarks on matrix exponential, Linear Algebra Appl., 117, 127-132 (1989), MR 90e:15019 · Zbl 0669.15009 [9] Wermuth, E. M.E., A remark on commuting operator exponentials, Proc. Amer. Math. Soc., 125, 6, 1685-1688 (1997), MR 97g:39011 · Zbl 0866.39004 [10] Wu, W., An order characterization of commutativity for \(C^\ast \)-algebras, Proc. Amer. Math. Soc., 129, 983-987 (2001), MR 2001j:46084 · Zbl 0968.46040 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.