Abstract
The classical theory of Sobolev towers allows for the construction of an infinite ascending chain of extrapolation spaces and an infinite descending chain of interpolation spaces associated with a given \(C_0\)-semigroup on a Banach space. In this note we first generalize the latter to the case of a strongly continuous and exponentially equicontinuous semigroup on a complete locally convex space. As a new concept—even for \(C_0\)-semigroups on Banach spaces—we then define a universal extrapolation space as the completion of the inductive limit of the ascending chain. Under mild assumptions we show that the semigroup extends to this space and that it is generated by an automorphism of the latter. Dually, we define a universal interpolation space as the projective limit of the descending chain. We show that the restriction of the initial semigroup to this space is again a semigroup and always has an automorphism as generator.
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Acknowledgments
The author likes to thank B. Jacob for several fruitful discussions on the topic of this note. Moreover, he likes to thank B. Farkas, who drew the author’s attention to the study of inductive and projective limits of Sobolev towers. Finally, the author likes to thank the referee for his careful work and his valuable comments.
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Wegner, SA. Universal extrapolation spaces for \(\hbox {C}_{0}\)-semigroups. Ann Univ Ferrara 60, 447–463 (2014). https://doi.org/10.1007/s11565-013-0189-5
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DOI: https://doi.org/10.1007/s11565-013-0189-5