Twisted Representations of Product Systems of C^*-Correspondences: Wold Decomposition and Unitary Extensions

We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of C^* C ∗ -correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple (σ , T₁, T₂, … , Tₙ) ( σ , T 1 , T 2 , … , T n ) , where each (σ ,Tᵢ) ( σ , T i ) is an isometric covariant representation of a C^* C ∗ -correspondence. We then introduce twisted and doubly twisted covariant representations of product systems. For doubly twisted isometric representations, we prove the existence of a Wold decomposition, recovering earlier results for doubly commuting representations as special cases. We further obtain explicit descriptions of the resulting Wold summands and develop concrete Fock-type models realizing each component. We present non-trivial examples of these families. Finally, we construct unitary extensions via a direct-limit procedure. As applications, we obtain unitary extensions for several previously studied classes of operator tuples, including doubly twisted, doubly non-commuting, and doubly commuting isometries, and for a special class of doubly twisted representations of a product system.