We prove that for a strongly continuous semigroup \(T\) on the Fréchet space \(\omega\) of all scalar sequences, its generator is
a continuous linear operator \(A\in L(\omega)\) and that, for any \(x\in\omega\) and \(t\geq 0\)
the series \(\exp(tA)x=\sum_{k=0}^\infty\frac{t^k}{k!}A^k x\) converges to \(T(t)x\).
This solves a problem posed by Conejero in [2]. Moreover, we improve recent results of Albanese, Bonet, and Ricker [1] about semigroups on strict projective limits of Banach spaces.
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References- A. Albanese, J. Bonet, W. Ricker, \(C_0\)-semigroups and mean ergodic operators in a class of Fréchet spaces , J. Math. Anal. Appl., 365, 2010, 142-157.
- J.A. Conejero, On the existence of transitive and topologically mixing semigroups , Bull. Belg. Math. Math. Soc. Simon Stevin 14, 2007, 463-471.
- L. Frerick, E. Jordá, T. Kalmes, J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces , J. Math. Anal. Appl., 412, 2014, 121-124.
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