Yakymenko D. Unitarization of representations of primitive posets of tame type.

The thesis is devoted to investigation of systems of subspaces of Hilbert spaces. We study a relation between brick n-tuples of subspaces of a finite dimensional linear space, and irreducible n-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive coefficients, of orthogonal projections onto these subspaces equals the identity operator. We prove that brick systems that correspond to representations of primitive poset of tame type can be unitarized with some character. For any brick quadruple of subspaces we describe sets of characters that admit an unitarization. For representations of E6 in continuous spectrum we describe sets of characters that admit an unitarization. The counterexamples of non unitarizable brick system of subspaces are constructed. The methods for constructing unitarizable systems of subspaces are considered. Key words: unitarization, orthoscalarity, systems of subspaces of Hilbert spaces, representations of partially ordered sets, representation theory, extended Dynkin diagrams.