Closed range and nonclosed range adjointable operators on Hilbert \(C^*\)-modules
Let \(E,F\) be Hilbert \(C^*\)-modules over the same \(C^*\)-algebra. The Banach space of all adjointable operators \(A:E\rightarrow F\) is denoted by \(L(E,F)\) (or \(L(E)\) when \(E=F\)). The range of an operator \(A \in L(E,F)\) is denoted by \(R(A)\) and \(\overline{R(A)}\) denotes its closure. The main result of the paper characterizes positive operators \(A\in L(E)\) with closed range. It is proved that a positive operator \(A\in L(E)\) has closed range if and only if \(R(A^{\alpha})\) is closed for some \(\alpha >0\) if and only if \(R(A^{\alpha})\) is closed for all \(\alpha>0\), and this occurs precisely when \(R(A)=R(A^{\alpha})\) for all \(\alpha>0\). In another theorem, the authors prove that \(A\in L(E,F)\) has closed range if and only if \(R(A)=R((AA^*)^{\alpha})\) for all \(\alpha >0\) (in addition, more equivalent conditions are found). In particular, the authors show by an example that this equality may fail for \(\alpha=\frac{1}{2}\) if the range of \(A\) is not closed. Finally, if \(A\in L(E,F)\) has the property that \(\overline{R(A^*)}\) is orthogonally complemented, a variant of a generalized inverse is obtained under certain additional condition.
scientific article; zbMATH DE number 6932289
scientific article