Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then $A+B$ generates an analytic $C_0$-semigroup on $X$ as well. If $B$ is even bounded from $X$ to $X$, then the Dyson–Phillips series representation $$e^{t(A+B)}=\sum_{n=0}^\infty S_k(t),\qquad t\ge 0,$$ where $$S_0(t):=e^{tA},\qquad S_{k+1}(t):=\int_0^t S_k(s)Be^{(t-s)A} ds$$ of the semigroup generated by $A+B$ holds (integrals converging in strong sense!).
What about the general case? Because $e^{tA}$ maps $X$ into $D(A)$ for all $t>0$, $S_k(s)Be^{(t-s)A}$ is a well-defined bounded linear operator on $X$ for all $0 \le s < t$. Does the above series representation prevail?
