Let \(A, B \in \mathbb{R}^{d \times d}\) and set \(P_{t}:=\exp (t A):=\sum_{j=0}^{\infty}(t A)^{j} / j\) !.

a) Show that \(P_{t}\) is a strongly continuous semigroup. Is it contractive?

b) Show that \(\frac{d}{d t} e^{t A}\) exists and that \(\frac{d}{d t} e^{t A}=A e^{t A}=e^{t A} A\).

c) Show that \(e^{t A} e^{t B}=e^{t B} e^{t A} \forall t \Longleftrightarrow A B=B A\). Differentiate \(e^{t A} e^{s B}\) at \(t=0\) and \(s=0\).

d) (Trotter product formula) Show that \(e^{A+B}=\lim _{k ightarrow \infty}\left(e^{A / k} e^{B / k}ight)^{k}\).