On Isometric Representations of the Perforated Semigroup

TL;DR

This paper investigates isometric representations of a specific semigroup, introduces inverse representations, and classifies irreducible non-inverse representations, advancing understanding of their structure and equivalence classes.

Contribution

It provides a complete classification of inverse isometric representations and explores a new class of non-inverse irreducible representations called β-representations.

Findings

Complete description of inverse representations up to unitary equivalence

Introduction and analysis of β-representations of the semigroup

Characterization of irreducible non-inverse representations

Abstract

We study isometric representations of the semigroup $\mathbb{Z}_+\backslash \{1\}$. Notion of an inverse representation is introduced and a complete description (up to unitary equivalence) of such representations is given. Also, we study a class of non-inverse irreducible representations -- $\beta$-representations of the semigroup $\mathbb{Z}_+\backslash \{1\}$.