If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute

Question

Let $A,B$ be complex valued square matrices. If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute.

The converse of this statement can be an easy application of the Cauchy product rule and the binomial theorem.

Note that this statement doesn't hold, if we restrict ourselves to $t = 1$.

So far I have been trying to use the fact, that $A$ and $B$ are infinitesimal generators to the semigroups $\{\exp(tA)\}$ and $\{\exp(tB)\}$ but I have had no success. Do you have any other hints?

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Based on the idea of @Did, I came up with the following:

Series expansions give me: $$ \sum_{n = 0}^\infty \frac{t^n(A + B)^n}{n!} = I + tA + tB + \frac{t^2(AB + BA)}{2} + \sum_{n = 3}^\infty \frac{t^n(A + B)^n}{n!} $$ and $$ \left(\sum_{n = 0}^\infty \frac{t^n(A)^n}{n!} \right) \left(\sum_{n = 0}^\infty \frac{t^n(B)^n}{n!} \right) = I + tA + tB + \frac{t^2A^2}{2} + t^2AB + \frac{t^2B^2}{2} + \sum_{n = 3}^\infty t^n c_n, $$ where $$ c_n := \sum_{k = 0}^n \frac{A^k B^{n - k}}{k! n!}. $$

The comparison of both expansions gives $$ \frac{t^2(AB + BA)}{2} + \sum_{n = 3}^\infty \frac{t^n(A + B)^n}{n!} = t^2AB + \sum_{n = 3}^\infty t^n c_n. $$ Division by $t > 0$ yields: $$ \frac{(AB + BA)}{2} + \sum_{n = 3}^\infty \frac{t^{n-2}(A + B)^n}{n!} = AB + \sum_{n = 3}^\infty t^{n-2} c_n. $$ But I can't quite see, why the two sums $\sum_{n = 3}^\infty \dots$ should go to $0$ for $t \to 0$ yielding the desired equality $$ \frac{(AB + BA)}{2} = AB . $$

Answer 1

Let us write $T(t)=e^{t(A+B)}$ and $S(t)=e^{tA}e^{tB}$. Then, using (for both) the product rule and (for $S(t)$) the fact that the generator commutes with the semigroup,

$$\frac{d}{dt}T(t)=(A+B)T(t),\quad \frac{d}{dt}S(t)=AS(t)+S(t)B$$ $$\frac{d^2}{dt^2}T(t)=(A+B)^2T(t),\quad \frac{d^2}{dt^2}S(t)=A^2S(t)+2AS(t)B+S(t)B^2\tag{1}$$

Since $T(t)=S(t)$ for all $t\geq 0$, we have $$\frac{d^2}{dt^2}T(t)=\frac{d^2}{dt^2}S(t),\quad\forall\ t\geq 0$$ and thus, from $(1)$, $$(AB+BA)S(t)+B^2S(t)=2AS(t)B+S(t)B^2,\quad\forall\ t\geq 0.$$ In particular, for $t=0$, $$AB+BA=2AB$$ and the desired result follows.

Remark: This solution follows the hint in Engel's book, page 23.