Universal extrapolation spaces for C$_0$-semigroups

TL;DR

This paper introduces universal extrapolation and interpolation spaces for $C_0$-semigroups, extending classical Sobolev tower concepts to more general locally convex spaces and establishing their properties.

Contribution

It generalizes Sobolev towers to locally convex spaces and defines universal extrapolation and interpolation spaces with extension and automorphism properties.

Findings

Universal extrapolation space is the completion of the inductive limit.

Semigroup extends to the universal extrapolation space as an automorphism.

Restriction to the universal interpolation space yields a semigroup with an automorphism generator.

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.