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We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariantrepresentation of a product system of $C^*$-correspondences. This extends arecent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commutingisometries. Application to the wandering subspaces of doubly commuting inducedrepresentations is explored, and a version of Mandrekar's Beurling type theoremis obtained to study doubly commuting invariant subspaces using Fock spaceapproach due to Popescu.

Doubly commuting invariant subspaces for representations of product systems of $C^*$-correspondences