On Isometric Representations of the Perforated Semigroup

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 $\mathbb{Z}+\backslash {1}$.