This talk revolves around tuples of isometries. We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We prove that each doubly twisted isometry admits a von Neumann-Wold type orthogonal decomposition. As an application of this decomposition, we study the universal C*-algebra and the irreducible representations of the C*-algebras generated by this family. Also, we exhibit concrete analytic models for this family. Next, we study orthogonal decompositions for general tuples of twisted isometries which also include the case of tuples of commuting isometries.

Finally, we consider contractive representations of the odometer semigroup. O_n, also known as the adding machine or the Baumslag-Solitar monoid with two generators, which is a well-known object of study in group theory. We examine the odometer semigroup in relation to representations of bounded linear operators. We focus on noncommutative operators and prove that contractive representations of O_n always admit to nicer isometric representations of O_n on vector-valued Fock spaces. We describe representations of O_n on the Fock space and relate it to the dilation theory and invariant subspaces of Fock representations of O_n. Along the way, we also classify Nica covariant representations of O_n.