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 \((k, \lambda )\).

中文翻译：

---

不确定内积空间上与加权投影相关的加权摩尔-彭罗斯逆

令H为Hilbert \（C ^ * \）-模，令\（H_M \）为H上可自邻接且可逆的算子M引起的不确定内部空间。给定\（H_M \）上的加权投影P和Q，让\（S _ {\ lambda，k} =（PQ）^ k- \ lambda（QP）^ k \）对\（（k，\ lambda） \），其中k是自然数，\（\ lambda \）是复数。证明，当且仅当对每对\（S _ {\ lambda，k} \）加权Moore-Penrose可逆时，\（PQ-QP \）才是加权Moore-Penrose可逆\（（k，\ lambda）\）。