Moore-Penrose inverses of Gram operators on Hilbert \(C^*\)-modules (Q2907270)

Let \(\mathcal {X}\) and \(\mathcal {Y}\) be two Hilbert \(\mathcal {A}\)-modules, where \(\mathcal {A}\) is an arbitrary \(C^*\)-algebra. An \(\mathcal {A}\)-linear operator \(t: \mathcal {X}\to \mathcal {Y}\) is said to be regular if \(t\) is densely defined, closed, adjointable, and the range of \(1+t^*t\) is dense in \(\mathcal {X}\). The set of all regular operators from \(\mathcal {X}\) to \(\mathcal {Y}\) is denoted by \(\mathcal {R}(\mathcal {X}, \mathcal {Y})\). For \(t\in \mathcal {R}(\mathcal {X}, \mathcal {Y})\), an operator \(s\in \mathcal {R}(\mathcal {Y}, \mathcal {X})\) is called a Moore-Penrose of \(t\) if \(tst=t\), \(sts=s\), \((ts)^*=\overline {ts}\), and \((st)^*=\overline {st}\). If \(t\) admits a unique Moore-Penrose inverse, then its Moore-Penrose inverse is denoted by \(t^{\dag }\). Let \(T\) be a bounded linear operator with closed range between two Hilbert spaces. We call \(T^*T\) the Gram operator of \(T\).NEWLINENEWLINEIn the paper under review, the authors study Moore-Penrose invertibility of the Gram operator \(t^*t\) by obtaining some conditions under which both NEWLINE\[NEWLINEt^{\dag }=(t^*t)^{\dag }t^*=t^*(tt^*)^{\dag }NEWLINE\]NEWLINE and NEWLINE\[NEWLINE(t^*t)^{\dag }=t^{\dag }t^{*\dag }NEWLINE\]NEWLINE hold true for a regular operator \(t\).