Intermediate and extrapolated spaces for bi-continuous semigroups

TL;DR

This paper develops a framework for constructing Sobolev and H"older scales for non-densely defined operators on Banach spaces, focusing on extrapolation and Favard spaces for bi-continuous semigroups.

Contribution

It introduces a novel method to construct Sobolev and H"older scales for generators of bi-continuous semigroups on Saks spaces, extending classical operator theory.

Findings

Constructed Sobolev and H"older scales for non-densely defined operators.

Provided a new approach for extrapolation and Favard spaces of semigroup generators.

Extended Hille-Yosida theory to Saks spaces.

Abstract

We discuss the construction of the full Sobolev (H\"older) scale for non-densely defined operators on a Banach space with rays of minimal growth. In particular, we give a construction for extrapolation- and Favard spaces of generators of (bi-continuous) semigroups, or which is essentially the same, Hille-Yosida operators on Saks spaces.