Lattice properties of the core-partial order
Let \({\mathcal B}(\mathcal H)\) denote the space of bounded linear operators on the Hilbert space \(\mathcal H\); then \(A \in {\mathcal B}(\mathcal H)\) is said to be \textit{group-invertible} if there exists \(X \in {\mathcal B}(\mathcal H)\) such that \(AXA=A\), \(XAX=X\) and \(AX=XA\). Such an operator, if it exists, is unique and is denoted by \(A^{\sharp}\) and is called the \textit{group inverse} of \(A\). The core-inverse of \(A \in {\mathcal B}(\mathcal H)\), if it exists, is the unique operator \(X \in {\mathcal B}(\mathcal H)\) that satisfies the equations: \(AXA=A\), \(XAX=X, (AX)^*=AX\) and \(AX^2=X\). The core-inverse of \(A\) is denoted by \(A^{\bigcirc\!\!\!\!{\sharp}}\). The core-partial order on \(\mathcal H\), denoted by \({\leq}^{\bigcirc\!\!\!\!{\sharp}}\), is defined by: \(A {\leq}^{\bigcirc\!\!\!\!{\sharp}}B\) if and only if \(A^{\bigcirc\!\!\!\!{\sharp}}A=A^{\bigcirc\!\!\!\!{\sharp}}B\) and \(AA^{\bigcirc\!\!\!\!{\sharp}}=BA^{\bigcirc\!\!\!\!{\sharp}}\). It is shown that the set of group-invertible operators on \(\mathcal H\) has the property that an arbitrary family of operators possesses the core-infimum property. Considerations over an arbitrary family are made with regard to the core-supremum, and other extensions are obtained.
scientific article; zbMATH DE number 6694361