On a certain class of semigroups of operators

TL;DR

This paper introduces a new class of operator semigroups called randomly generated semigroups in Banach spaces, linking them to group representations and convolution semigroups, with applications in physics.

Contribution

It defines and explores the properties of randomly generated semigroups, connecting them to quantum dynamical semigroups and group theory, expanding the framework for operator semigroups.

Findings

Identification of randomly generated semigroups as a broad class including quantum dynamical semigroups

Connection established between semigroups, group representations, and convolution measures

Examples provided demonstrating applications in physics

Abstract

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced by Kossakowski in the early 1970s. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.