Open problems in one-parameter operator semigroups theory

TL;DR

This paper compiles unresolved open problems in the theory of one-parameter operator semigroups, highlighting current challenges and research directions discussed at recent international conferences.

Contribution

It presents a curated list of unsolved open problems in semigroup theory from recent OPSO conferences, with context and relevance for ongoing research.

Findings

Collection of open problems from 2021-2024 OPSO conferences

Comments on the relevance and history of each problem

Identification of key challenges in semigroup theory

Abstract

One-parameter strongly continuous semigroups of linear bounded operators on Banach spaces (also known as $C_0$-semigroups) are a fundamental operator-theoretic tool used in the study of linear and non-linear evolution PDEs arising in physics, probability, control theory and other areas of science and technology, including quantum theory, transportation problems and finance. Since 2021 the annual online conference on one-parameter semigroups of operators (OPSO) offers an opportunity for researchers worldwide to discuss the current state of the art and exchange knowledge. On every OPSO conference open problems are proposed, collected and discussed, some of them are found to be solved and hence get crossed out from the list. In this paper we provide a collection of open problems of semigroup theory that were proposed by participants of OPSO 2021--2024 conferences and have not been solved, yet. For each problem some comments on the relevance and and history of the problem are provided.