Compact perturbations of operator semigroups

We study lifting problems for operator semigroups in the Calkin algebra $\mathscr{Q}(\mathcal{H})$, our approach being mainly based on the Brown--Douglas--Fillmore theory. With any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathscr{Q}(\mathcal{H})$ we associate an extension $\Gamma\in\mathrm{Ext}(\Delta)$, where $\Delta$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $\sigma(A)$ of the generator of $(q(t))_{t\geq 0}$. By using Milnor's exact sequence, we show that if each $q(t)$ has a normal lift, then the question whether $\Gamma$ is trivial reduces to the question whether the corresponding first derived functor vanishes. With the aid of the CRISP property and Kasparov's Technical Theorem, we provide geometric conditions on $\sigma(A)$ which guarantee splitting of $\Gamma$. If $\Delta$ is a perfect compact metric space, we obtain in this way a $C_0$-semigroup $(Q(t))_{t\geq 0}$ which lifts $(q(t))_{t\geq 0}$ on dyadic rationals.