Strongly continuous semigroups on some Fréchet spaces

We prove that for a strongly continuous semigroup T on the Fréchet space ω of all scalar sequences, its generator is a continuous linear operator A∈L(ω) and that, for all x∈ω and t⩾0, the series exp(tA)(x)=∑k=0∞tkk!Ak(x) converges to Tt(x). This solves a problem posed by Conejero. Moreover, we improve recent results of Albanese, Bonet, and Ricker about semigroups on strict projective limits of Banach spaces.