Automatic continuity of operator semigroups in the Calkin algebra

We study operator semigroups in the Calkin algebra Q(H), represented as a subalgebra of the algebra of bounded linear operators on a Hilbert space via one of `canonical' Calkin's representations. Using the BDF theory, we associate with any normal C₀-semigroup (q(t))t≥ 0 in Q(H) an extension Γ(Δ), where Δ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum σ(A) of the generator of (q(t))t≥ 0. Then we show that, in natural circumstances, if (q(t))t≥ 0 is continuous in the strong operator topology, then it is actually uniformly continuous, although there are C₀-semigroups in Q(H) that are not uniformly continuous.