Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2020-07-14 , DOI: 10.1007/s41980-020-00432-3 Guanjie Yan , Yunfei Tan , Qingxiang Xu
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\((k,\ lambda)\)。