An elementary approach to splittings of unbounded operators

Abstract

Using elementary means, we derive the three most popular splittings of $e^{t(A+B)}$ and their error bounds in the case when A and B are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups, $e^{tA}$, $e^{tB}$ and $e^{t(A+B)}$. The error of these splittings is bounded in terms of the norm of the commutators $[A,B]$, $[A,[A,B\left]\right]$ and $[B,[A,B\left]\right]$.