Slice regular semigroups

TL;DR

This paper introduces slice regular right linear semigroups in quaternionic Banach spaces, establishing a link between slice regularity and noncommutative semigroup theory, and generalizing results to modules over real associative *-algebras.

Contribution

It defines slice regular semigroups in quaternionic Banach spaces and characterizes them via spherical sectorial generators, extending to modules over real *-algebras.

Findings

Slice regular semigroups characterized by spherical sectorial generators

Connection established between slice regularity and noncommutative semigroup theory

Generalization to Banach modules over real *-algebras

Abstract

In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy integral formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford R_n algebras included.