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) ) ₓ ₀ in Q (H) an extension (), where is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum (A) of the generator of (q (t) ) ₓ ₀. Then we show that, in natural circumstances, if (q (t) ) ₓ 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.