A Short Counterexample to the Inverse Generator Problem on non-Hilbertian Reflexive $L^p$-spaces

TL;DR

This paper presents a counterexample to the inverse generator problem using the bilateral shift group on non-Hilbertian reflexive L^p-spaces, specifically for p in (1, ∞) excluding 2.

Contribution

It demonstrates that the bilateral shift group on L^p(ℝ) for p ≠ 2 is a counterexample to the inverse generator problem, highlighting limitations in the theory.

Findings

Counterexample on L^p(ℝ) for p ≠ 2

Shows bilateral shift group does not satisfy inverse generator property

Highlights differences between Hilbert and non-Hilbert spaces

Abstract

We show that the bilateral shift group on $L^p(\mathbb{R})$ for $p \in (1, \infty) \setminus \{2\}$ provides a counterexample to the inverse generator problem.