Embedding operators into strongly continuous semigroups

TL;DR

This paper investigates conditions under which linear operators on Banach spaces can be embedded into strongly continuous semigroups, focusing on spectral properties and providing examples of possible and impossible embeddings.

Contribution

It introduces a necessary spectral condition for embedding operators into $C_0$-semigroups and explores classes of operators where embedding is feasible or not.

Findings

Spectral value 0 is crucial for embedding.

Examples of operators that can be embedded.

Examples of operators that cannot be embedded.

Abstract

We study linear operators $T$ on Banach spaces for which there exists a $C_0$-semigroup $(T(t))_{t\geq 0}$ such that $T=T(1)$. We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or cannot be achieved.