Representations for weighted Moore-Penrose inverses of partitioned adjointable operators

Abstract

For two positive definite adjointable operators $M$ and $N$, and an adjointable operator $A$ acting on a Hilbert $C^{\ast }$-module, some properties of the weighted Moore-Penrose inverse $A_{MN}^{†}$ are established. When $A=(A_{ij})$ is $1\times 2$ or $2\times 2$ partitioned, general representations for $A_{MN}^{†}$ in terms of the individual blocks $A_{ij}$ are studied. In case $A$ is $1\times 2$ partitioned, a unified representation for $A_{MN}^{†}$ is presented. In the $2\times 2$ partitioned case, an approach to constructing Moore-Penrose inverses from the non-weighted case to the weighted case is provided. Some results known for matrices are generalized in the general setting of Hilbert $C^{\ast }$-module operators.

arXiv commentAccepted for publication in Linear Algebra and its Applications. 20 pages