Yusenko A. Structural theorems for the sets of operators with given spectrums

The thesis is devoted to the subsequent investigation of the collections of those orthoprojections in some Hilbert space, that the linear combination of which is equal to the scalar operator; the set of those parameters for which there exist representation of linear relation in the category of Hilbert spaces and we build irreducible representations of this relation. Namely in the case when n=4 we described all irreducible up to the unitary equivalence collections of four orthoprojections in certain Hilbert space, such that their linear combination is equal to scalar operator. In the case when n is more 4, we showed that the set of parameters, for which there exist representation of the linear relation, contains nonempty open set. Also we found sufficient condition on the sum of coefficients of those parameters for which there always exist representations of linear relation.