Projections and Idempotents in star-reducing Rings Involving the Moore-Penrose Inverse

Abstract

In [Comput. Math. Appl. 59 (2010) 764-778], Baksalary and Trenkler characterized some complex idempotent matrices of the form $(I-PQ{)}^{†}P(I-Q)$, $(I-QP{)}^{†}(I-Q)$ and $P(P+Q-QP{)}^{†}$ in terms of the column spaces and null spaces of $P$ and $Q$, where $P,Q\in {\mathbb{C}}_n^{\text{OP}}=\{L\in {\mathbb{C}}_{n,n}|L^2=L=L^{\ast }\}$. We generalize these results from ${\mathbb{C}}_{n,n}$ to any *-reducing rings.

17 pages