# Dudkin M. Ye.

### Jacobi-type block matrices corresponding to the two-dimensional moment problem: polynomials of the second kind and Weyl function

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 495-505

We continue our investigations of Jacobi-type symmetric matrices corresponding to the two-dimensional real power moment problem. We introduce polynomials of second kind and the corresponding analog of the Weyl function.

### Point spectrum of the schrödinger operator with point interactions at the vertices of regular *N*-gons

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1128–1134

We present a complete description of the point spectrum of the Laplace operator perturbed by point potentials concentrated at the vertices of regular polygons. We prove a criterion for the absence of points of the point spectrum of a singular perturbed positive self-adjoint operator with the property of cyclicity of defect vectors.

### On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

Dudkin M. Ye., Koshmanenko V. D.

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1269-1276

We discuss purely singular finite-rank perturbations of a self-adjoint operator *A* in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im *z* ≠ 0, where *B* _{z} are finite-rank operators such that dom *B* _{z} ∩ dom *A* = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ_{ i }} ∩ dom *A* = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank *B* _{z} = *n*.

### Analog of the Krein Formula for Resolvents of Normal Extensions of a Prenormal Operator

Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 555-562

We prove a formula that relates resolvents of normal operators that are extensions of a certain prenormal operator. This formula is an analog of the Krein formula for resolvents of self-adjoint extensions of a symmetric operator. We describe properties of the defect subspaces of a prenormal operator.

### Singularly perturbed normal operators

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1045–1053

We present a generalization of definition'of-singularly perturbed operators to the case of normal operators. To do this, we use an idea of normal expansions of a prenormal operator and prove the relation for resolvents of normal expansions similar to the M. Krein relation for resolvents of self-adjoint expansions. In addition, we establish one-to-one correspondence between the set of singular perturbations of rank one and the set of perturbed (of rank one) operators.

### Invariant symmetric restrictions of a self-adjoint operator. II

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 781–791

We give a criterion of invariance and symmetry of the restriction of an arbitrary unbounded self-adjoint operator in the space *L* _{2}(ℝ^{n}, *dx*) by using the introduced notion of support of an arbitrary vector and the notion of capacity of a subspace *N* ⊂ ℝ^{n}.

### Invariant symmetric restrictions of a self-adjoint operator. I

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 623–631

We prove necessary and sufficient conditions of the *S*-invariance of a subset dense in a separable Hilbert space *H*.