Let \(\left(P_{t}ight)_{t \geqslant 0}\) and \(\left(T_{t}ight)_{t \geqslant 0}\) be two Feller semigroups with generators \((A, \mathfrak{D}(A))\), resp. \((B, \mathfrak{D}(B))\).

a) If \(\mathfrak{D}(B) \subset \mathfrak{D}(A)\), then we have \(\frac{d}{d s} P_{t-s} T_{s}=-P_{t-s} A T_{s}+P_{t-s} B T_{s}\) on \(\mathfrak{D}(B)\).

b) Is it possible that \(U_{t}:=T_{t} P_{t}\) is again a Feller semigroup?

c) Assume that \(U_{t} f:=\lim _{n ightarrow \infty}\left(T_{t / n} P_{t / n}ight)^{n} f\) exists strongly and locally uniformly for \(t\). Show that \(U_{t}\) is a Feller semigroup and determine its generator.