Weighted Moore–Penrose Inverses Associated with Weighted Projections on Indefinite Inner Product Spaces

Abstract

Let H be a Hilbert $C^*$-module, and let $H_M$ be the indefinite inner space induced by a self-adjointable and invertible operator M on H. Given weighted projections P and Q on $H_M$, let $S_{\lambda ,k}=(PQ)^k-\lambda (QP)^k$ for a pair $(k, \lambda )$, where k is a natural number and $\lambda$ is a complex number. It is proved that $PQ-QP$ is weighted Moore–Penrose invertible if and only if $S_{\lambda ,k}$ is weighted Moore–Penrose invertible for every pair $(k, \lambda )$.