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 ( t A ) ( x ) = ∑ k = 0 ∞ t k k ! A k ( x ) converges to T t ( 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.