Wold-type decomposition for doubly twisted left-invertible covariant representations

In this article, we have introduced the notion of a near-isometric covariant representation of a $C^*$-correspondence. The other objective is to provide a unified approach to several known results for a large class of left-invertible covariant representations of a product system and prove Wold-type decomposition for the case of doubly twisted left-invertible covariant representations and study some applications.