Cauchy dual and Wold-type decomposition for bi-regular covariant representations

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for {regular} completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation $(\sigma, V)$ modulo $N(\wV)$ is hyponormal modulo $N(\wV)$.