Small transitive families of subspaces in finite dimensions (Q1855419)

Let \(\mathcal{F}\) be a family of norm-closed subspaces of the complex Hilbert space \(H\) and \(\text{Alg\,} \mathcal{F}\) denote the algebra of all bounded linear operators on \(H\) which leave every element of \(\mathcal{F}\) left invariant. The family \(\mathcal{F}\) is said to be transitive if every element of \(\text{Alg\,} \mathcal{F}\) is a scalar multiple of the identity operator. In the paper under review, the authors are interested in transitive families of minimum cardinality on finite-dimensional spaces. They show that on a complex finite-dimensional Hilbert space of dimension at least \(3\), the minimum cardinality of a transitive family of subspaces is \(4\). If \(\dim H=2\) , the minimum cardinality of a transitive family of subspaces is \(3\). They describe all \(4\)-element transitive family of subspaces of \(3\)-dimensional space. They obtain necessary, but not sufficient conditions satisfied by every \(4\)-element transitive family for spaces of dimension greater than \(3\).