Dichotomy results for norm estimates in operator semigroups

TL;DR

This survey explores how the behavior of operator semigroups at the origin influences their properties like continuity and analyticity, highlighting classical and recent results in the field.

Contribution

It provides a comprehensive overview of dichotomy results linking the initial behavior of semigroups to their qualitative properties, including new insights from recent research.

Findings

Semigroups exhibit '0-1' laws connecting initial behavior to properties.

Classical theorems of Beurling, Kato, and Neuberger are discussed.

Recent work extends these dichotomy results.

Abstract

This is a survey paper concerned with strongly continuous semigroups in a Banach algebra (often itself simply the algebra of bounded linear operators on a Banach space). These are defined either on $(0,\infty)$ or on a sector in the complex plane, in which case they are supposed analytic. However, the semigroups are not assumed to be strongly continuous at 0. The results in this survey, beginning with classical theorems of Beurling, Kato and Neuberger and concluding with recent work by the authors, are of the nature of "0-1" laws, indicating that the quantitative behaviour of the semigroup at the origin provides additional qualitative information, such as uniform continuity or analyticity.