Weighted inner weak group inverse

Solving a system of operator equations which extends early studied systems of equations, we present the notion of weighted inner weak group inverse for bounded linear operators between two Hilbert spaces. In this way, we generalize the concepts of the weighted MP weak group inverse for rectangular matrices and the 1WG inverse for square matrices. We give many characterizations and expressions for weighted inner weak group inverse. The operator matrix form of weighted inner weak group inverse is established. We consider the inner weak group inverse of operators as a special case of the weighted inner weak group inverse. The weighted inner weak group inverse is applied in solving certain linear equations and in representing their general solution forms.