Reverse order laws for the Drazin inverses (Q530350)

Let \({\mathcal B}({\mathcal H})\) denote the space of all bounded linear operators on a complex Hilbert space \({\mathcal H}\). Let \(T \in {\mathcal B}({\mathcal H})\). If there exists \(X \in {\mathcal B}({\mathcal H})\) such that \(TX=XT, ~XTX=X\) and \(T^{K+1}X=T^k\), where \(k\) denotes the index of \(T\), then such an \(X\) is unique and is called the \textit{Drazin inverse} of \(T\). In such a case, \(T\) is said to be \textit{Drazin invertible} and the Drazin inverse is denoted by \(T^D\). The authors study the problem of when the following ``reverse order law'' holds for Drazin invertible operators \(P\) and \(Q\): \((PQ)^D=Q^DP^D\). Many characterizations are proved and here is a sample: Theorem. Let \([A,B]:=AB-BA\) be the commutator of \(A\) and \(B\). Suppose that \([P,PQ]=0\). Then the reverse order law holds if and only if \((PQ)^DP=Q^DP^DP\).