Infima of Hilbert space effects
If \(H\) is a Hilbert space let \({\mathcal E}(H)\) denote the set of positive operators on \(H\) that are bounded above by the unit operator. \({\mathcal E}(H)\) is a partially ordered set which fails to be a lattice even if \(H\) is finite-dimensional, i.e., \(H= \mathbb{C}^n\). The authors investigate conditions under which the infimum \(A\wedge B\) of two elements \(A,B\in{\mathcal E}(H)\) exists. If \(A\in{\mathcal E}(H)\) let \(R(A)\) be the closure of the range of \(A\) and for \(A,B\in{\mathcal E}(H)\) let \(P_{A,B}\) denote the projection to \(R(A)\cap R(B)\). Using methods of linear algebra the authors show that for \(A,B\in{\mathcal E}(\mathbb{C}^n)\), \(n\in\mathbb{N}\), \(A\wedge B\) exists if and only if \(A\wedge P_{A,B}\) and \(B\wedge P_{A,B}\) are comparable in which case \(A\wedge B\) is the smaller of them.
scientific article; zbMATH DE number 1331722