Dudkin N. Syngularly perturbed normal operators and the complex moment problem.

The thesis is devoted to the investigation of the complex moment problem and related questions of the spectral theory of normal operators. Based on the eigenfunction expansion, there are proved the necessary and sufficient condition for a solvability of the complex moment problem in the power and exponential forms and there are described conditions of their uniqueness solvability. For unitary and normal operators there are described block threediagonal matrix Jacobi corresponding to these problems. There are proved the necessary and sufficient for existence of singularly perturbed normal operators and there is obtained the complete description of singularly perturbed rank one normal operators using introduced concept of admissible vector and the rigged Hilbert spaces. There is given the construction of the singularly perturbed normal operator with given eigenvectors and eigenvalues. There is investigated the point spectrum of the Laplace operator perturbed by delta-interactions concentrated on the top of some rectilinear geometric figures in R3. Using the concept of the capacity in the abstract Hilbert space there is given the condition of sufficient normality of the prenormal restriction of a normal operator.