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 A(MN)(dagger) are established. If A = (A(ij)) is 1 x 2 or 2 x 2 partitioned, then general representations for AN in terms of the individual blocks of A(ij) are studied. In the case when A is 1 x 2 partitioned, a unified representation for A(MN)(dagger) is presented. In the 2 x 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. (C) 2012 Elsevier Inc. All rights reserved.