The $m$-weak group orthogonality for operators

The main goal is extending the concept of the core-EP orthogonality to the $m$-weak group orthogonality for bounded linear Drazin invertible Hilbert space operators, using the $m$-weak group inverse. Different properties and characterizations of $m$-weak group orthogonal operators are proved as well as their operator matrix forms. The connection between the $m$-weak group binary relation and the $m$-weak group orthogonality is given. We also study additive properties for the $m$-weak group inverse. Consequently, we study the weak group orthogonality for operators.