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 $$(k, \lambda )$$ .