Grushevoi R. On systems of subspaces of a Hilbert space

Investigations of n-tuples of subspaces of linear space becomes a classical problem of linear algebra. Obtained results in this area have led to a solving of important mathematical problems. The thesis devoted to studying of systems of subspaces of a Hilbert space. Existence of irreducible configurations of one-dimensional subspaces of a Hilbert space associated with simple non-oriented graph Г and one angle т is investigated. All irreducible symmetric systems of one-dimensional subspaces up to unitary equivalence are described. Notion of Gram matrix for system of subspaces of a Hilbert space was proposed in the thesis. Using this notion some properties of a Coxeter functors and of systems of subspaces which obtained from one-dimensional ones using this functors are established. The sets of parameters v for which there exists decomposition of the operator vI into a sum of n self-adjoint operators having their spectra in the sets using orthoscalar systems of subspaces are described. In the thesis was proved that each indecomposable representation of poset of finite type could be unitarized with some weight and in case of primitive posets for each representation all appropriated for unitarization characters were described.