### 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.

Original language | English |
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Pages (from-to) | 447-463 |

Number of pages | 17 |

Journal | Annali dell'Universita di Ferrara |

Volume | 60 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 2014 |

### Keywords

- Extrapolation space
- Interpolation space
- Semigroup
- Sobolev tower

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

_{0}-semigroups.

*Annali dell'Universita di Ferrara*,

*60*(2), 447-463. https://doi.org/10.1007/s11565-013-0189-5