Session 17. Functional Analysis: relations to Complex Analysis and PDE

Strongly continuous semigroups on some Fréchet spaces

Thomas Kalmes, TU Chemnitz, 
The talk is based on the joint work with L. Frerick, E. Jordá, and J. Wengenroth.
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.
References
  1. 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.
  2. J.A. Conejero, On the existence of transitive and topologically mixing semigroups , Bull. Belg. Math. Math. Soc. Simon Stevin 14, 2007, 463-471.
  3. 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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