Representations for weighted Moore-Penrose inverses of partitioned adjointable operators

For two positive definite adjointable operators M and N, and an adjointable operator A acting on a Hilbert C∗-module, some properties of the weighted Moore-Penrose inverse AMN† are established. If A=(Aij) is 1 × 2 or 2 × 2 partitioned, then general representations for AMN† in terms of the individual blocks of Aij are studied. In the case when A is 1 × 2 partitioned, a unified representation for AMN† is presented. In the 2 × 2 partitioned case, an approach to the construction of the Moore-Penrose inverse from the non-weighted case to the weighted case is provided. Some results known for matrices are extended to the general setting of operators on Hilbert C∗-modules.