Doubly commuting invariant subspaces for representations of product systems of \(C^\)-correspondences

We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system of \(C^*\)-correspondences. This extends a recent Wold-type decomposition by Jeu and Pinto for a \(q\)-doubly commuting isometries. Application to the wandering subspaces of doubly commuting induced representations is explored, and a version of Mandrekar's Beurling type theorem is obtained to study doubly commuting invariant subspaces using Fock space approach due to Popescu.