\(\Gamma\)-inverses of bounded linear operators
Let \(\mathcal H\) be a Hilbert space and \(A,P,Q\) be bounded linear operators on \(\mathcal H\). If there exists a bounded linear operator \(X\) on \(\mathcal H\) such that \(APXQA=A\), \(XQAPX=X\), \((QAPX)^*=QAPX\) and \((XQAP)^*=XQAP\), then \(X\) is called the \(\Gamma\)-inverse of \(A\) associated with \(P\) and \(Q\) and is denoted by \(A^{\dagger}_{P,Q}\). If \(P=Q=I\), then \(A^{\dagger}_{P,Q}\) reduces to the Moore-Penrose inverse \(A^{\dagger}\). The main results of the authors are a necessary and sufficient condition for the existence of \(A^{\dagger}_{P,Q}\) and an explicit formula for it, when it exists.
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