A Characterization of Invariant Subspaces for Isometric Representations of Product System over \(\mathbb {N}_0^{k}\)

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over \(\mathbb {N}_0^{k}.\) ${\mathbb{N}}_0^k.$ As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system.