Reverse order law for the group inverses (Q549803)

The group inverse \(T^\#\) of an operator \(T\in B(H)\) is the unique element (if it exists) \(T^\#\in B(H)\) such that \(TT^\#=T^\#T, T^\#TT^\#=T^\#\) and \(T=TT^\#T\). Some necessary and sufficient conditions for the existence of group inverses for square matrices are given in [\textit{C.-G. Cao} and \textit{J.-Y. Li}, Electron. J. Linear Algebra 18, 600--612 (2009; Zbl 1189.15005)] and references therein. In the paper under review, the author obtains some equivalent conditions concerning the reverse order law \((TS)^\#=S^\#T^\#\) and shows that \((TS)^\#=S^\#T^\#\) if and only if \((TS)^\#T=S^\#T^\#T\) and the commutator equation \([I-T^\pi,ST^\pi]=0\) holds, where \(T^\pi=I-TT^\#\) is the spectral idempotent.