Small transitive families of subspaces in finite dimensions Original Research Article

A family image of subspaces of a finite-dimensional Hilbert spaceH is transitive if every operator leaving every element of image invariant is scalar. If dim Hgreater-or-equal, slanted3, the minimum cardinality of a transitive family is 4. All 4-element transitive families of subspaces of 3-dimensional space are described. For spaces of dimension greater than 3, necessary, but not sufficient, conditions satisfied by every 4-element transitive family are obtained, showing that (i) either every pair of subspaces intersects in (0) or every pair spans H (but not both), (ii) at least three of the subspaces must have the same dimension (either [dimH/2] or [dimH/2]+1), the dimension of the remaining subspace differing from this common dimension by at most 1.