$\mathbf {The \ Problem \ is}:$ If $A$ and $B$ are two bdd linear operators on a Banach space, then show that
$\lim_{n \to \infty} (e^\frac{A}{\sqrt n}e^\frac{B}{\sqrt n}e^\frac{-A}{\sqrt n}e^\frac{-B}{\sqrt n})^n = e^{(AB-BA)}$
$\mathbf {My \ approach}:$ Actually, we know $e^\frac{T}{\sqrt n} = I + \frac{T}{\sqrt n} + o(\frac{1}{\sqrt n}) $ where $T$ is a bdd linear operator and $o$ is Landau's little order notation , but the multiplication is getting bigger and confusing .
Is there any other better way to prove this thing except this multiplication .
A small hint is warmly appreciated .