### Abstract

It is a widely used method, for instance in perturbation theory, to associate with a given C_{0}-semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L^{2}(R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff and already complete if we consider a C_{0}-semigroup acting on a reflexive Banach space. If the space is Hilbert, then the inductive limit can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the latter duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space.

Original language | English |
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Pages (from-to) | 321-331 |

Number of pages | 11 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 460 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 1 2018 |

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### Keywords

- Dual with respect to a pivot space
- Extrapolation space
- Interpolation space

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*460*(1), 321-331. https://doi.org/10.1016/j.jmaa.2017.11.042