Twisted representations of product systems of $C^*$-correspondences: Wold decomposition and unitary extensions

Abstract: We investigate Wold-type decompositions and unitary extension problems for multivariable isometric covariant representations associated with product systems of $C^*$-correspondences. First, we establish an operator-theoretic characterization for the existence of a Wold decomposition for the tuple $(σ, T_1, T_2, \ldots, T_n)$, where each $(σ,T_i)$ is an isometric covariant representation of a $C^*$\nobreakdash-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 product system.