# Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE

$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$

which is just given by $$e^{t(A+B)}u_0$$ by the splitting $$e^{tB/2} e^{tA} e^{tB/2}u_0.$$ Here, $$A,B$$ can be assumed to be self-adjoint matrices.

We shall furthermore assume that $$A+B$$ generates a contraction semigroup, i.e. $$\Vert e^{t(A+B)} \Vert \le 1,$$ i.e. all eigenvalues of $$A+B$$ are negative.

Since structure preserving schemes are important, I am looking for a criterion such that $$\Vert e^{tB/2} e^{tA} e^{tB/2}\Vert \le C \text{ for all }t \in [0,\infty).$$

There are obviously some conditions like $$A,B$$ individually having only non-positive eigenvalues etc..., that would do the job, but I am looking for a more intelligent criterion that takes into account the fact that this is really supposed to be an approximation of the exponential of $$A+B$$.