PROPERTY L AND COMMUTING EXPONENTIALS IN DIMENSION AT MOST THREE

Abstract

AbstractLet$A, B$be two square complex matrices of the same dimension$n\leq 3$. We show that the following conditions are equivalent. (i) There exists a finite subset$U\subset { \mathbb{N} }_{\geq 2} $such that for every$t\in \mathbb{N} \setminus U$,$\exp (tA+ B)= \exp (tA)\exp (B)= \exp (B)\exp (tA)$. (ii) The pair$(A, B)$has property L of Motzkin and Taussky and$\exp (A+ B)= \exp (A)\exp (B)= \exp (B)\exp (A)$. We also characterise the pairs of real matrices$(A, B)$of dimension three, that satisfy the previous conditions.