Systems of subspaces of a unitary space

For a finite poset P={p1,…,pt}, we study systems (U1,…,Ut)U of subspaces of a unitary space U such that Ui⊆Uj if pi≺pj. Two systems (U1,…,Ut)U and (V1,…,Vt)V are said to be isometric if there exists an isometry φ:U→V such that φ(Ui)=Vi. We classify such systems up to isometry if P is a semichain. We prove that the problem of their classification is unitarily wild if P is not a semichain. A classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space, which is hopeless in a certain sense.