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 over N 0 k . This result gives Shimorin-type decompositions of recent Wold-type decompositions by Jeu and Pinto (Adv Math 368:107–149, 2020) for the q -doubly commuting isometries and by Popescu (J Funct Anal 279:108798, 2020) for Doubly Λ -commuting row isometries. Application to the wandering subspaces of the induced representations is explored, and a version of the Beurling–Lax-type characterization is obtained to study doubly commuting invariant subspaces.