Representations of partially ordered sets in Hilbert spaces. Unitarization

Abstract:
Let H be a separable Hilbert space. The system S =(H;H₁,...,H_n) of subspaces H_i of the space H is a representation of some finite partially ordered set (poset) P in Hilbert space H if the inclusion between the subspaces corresponds to the partial order in P and the linear combination of the projections on corresponding subspaces is equal to scalar operator. The representation theory of the posets in Hilbert space has deep interconnections with the representation theory of different object. For the case when P is primitive and of finite type we investigate when linear representation can present Hilbert representation.