# M. E. Dudkin

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### On nonsymmetric rank one singular perturbations of selfadjoint operators

Mykola Dudkin, Tetiana Vdovenko

Methods Funct. Anal. Topology **22** (2016), no. 2, 137-151

We consider nonsymmetric rank one singular perturbations of a selfadjoint operator, i.e., an expression of the form $\tilde A=A+\alpha\left\langle\cdot,\omega_1\right\rangle\omega_2$, $\omega_1\not=\omega_2$, $\alpha\in{\mathbb C}$, in a general case $\omega_1,\omega_2\in{\mathcal H}_{-2}$. Using a constructive description of the perturbed operator $\tilde A$, we investigate some spectral and approximations properties of $\tilde A$. The wave operators corresponding to the couple $A$, $\tilde A$ and a series of examples are also presented.

### Nonzero capacity sets and dense subspaces in scales of Sobolev spaces

Mykola E. Dudkin, Volodymyr D. Koshmanenko

Methods Funct. Anal. Topology **20** (2014), no. 3, 213-218

We show that for a compact set $K\subset{\mathbb R}^n$ of nonzero $\alpha$-capacity, $C_\alpha(K)>0$, $\alpha\geq 1$, the subspace $\overset{\circ}{W}{^{\alpha,2}}(\Omega)$, $\Omega={\mathbb R}^n\setminus K$ in ${W}{^{\alpha,2}}({\mathbb R}^n)$ is dense in $W^{m,2}({\mathbb R}^n)$, $m\leq\alpha-1$, iff the $m$-capacity of $K$ is zero, $C_{m}(K)=0$.

### Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem

Mykola E. Dudkin, Valentyna I. Kozak

Methods Funct. Anal. Topology **20** (2014), no. 3, 219-251

We generalize the connection between the classical power moment problem and the spectral theory of selfadjoint Jacobi matrices. In this article we propose an analog of Jacobi matrices related to some system of orthonormal polynomials with respect to the measure on the real plane. In our case we obtained two matrices that have a block three-diagonal structure and are symmetric operators acting in the space of $l_2$ type. With this connection we prove the one-to-one correspondence between such measures defined on the real plane and two block three-diagonal Jacobi type symmetric matrices. For the simplicity we investigate in this article only bounded symmetric operators. From the point of view of the two dimensional moment problem this restriction means that the measure in the moment representation (or the measure, connected with orthonormal polynomials) has compact support.

### The inner structure of the Jacobi-Laurent matrix related to the strong Hamburger moment problem

Methods Funct. Anal. Topology **19** (2013), no. 2, 97-107

The form of the Jacobi type matrix related to the strong Hamburger moment problem is known \cite{N5,BD}, i.e., there are known the zero elements of corresponding matrix. We describe the relations between of non-zero elements of such matrices, i.e., we describe ''the inner structure'' of the Jacobi-Laurent matrices related to the strong Hamburger moment problem.

### The complex moment problem in the exponential form with direct and inverse spectral problems for the block Jacobi type correspondence matrices

Methods Funct. Anal. Topology **18** (2012), no. 2, 111-139

We present a new generalization of the connection of the classical power moment problem with spectral theory of Jacobi matrices. In the article we propose an analog of Jacobi matrices related to the complex moment problem in the case of exponential form and to the system of orthonormal polynomials with respect to some measure with the compact support on the complex plane. In our case we obtain two matrices that have block three-diagonal structure and acting in the space of $l_2$ type as commuting self-adjoint and unitary operators. With this connection we prove the one-to-one correspondence between the measures defined on a compact set in the complex plane and the couple of block three-diagonal Jacobi type matrices. For simplicity we consider in this article only a bounded self-adjoint operator.

### Singularly perturbed normal operators

Methods Funct. Anal. Topology **16** (2010), no. 4, 298-303

We give an effective description of finite rank singular perturbations of a normal operator by using the concepts we introduce of an admissible subspace and corresponding admissible operators. We give a description of rank one singular perturbations in terms of a scale of Hilbert spaces, which is constructed from the unperturbed operator.

### The strong Hamburger moment problem and related direct and inverse spectral problems for block Jacobi-Laurent matrices

Yurij M. Berezansky, Mykola E. Dudkin

Methods Funct. Anal. Topology **16** (2010), no. 3, 203-241

In this article we propose an approach to the strong Hamburger moment problem based on the theory of generalized eigenvectors expansion for a selfadjoint operator. Such an approach to another type of moment problems was given in our works earlier, but for strong Hamburger moment problem it is new. We get a sufficiently complete account of the theory of such a problem, including the spectral theory of block Jacobi-Laurent matrices.

### An exact inner structure of the block Jacobi-type unitary matrices connected with the corresponding direct and inverse spectral problems

Methods Funct. Anal. Topology **14** (2008), no. 2, 168-176

We discuss a problem posed by M. J. Cantero, L. Moral, and L.~Vel\'azquez about representing an arbitrary unitary operator with a CMV-matrix. We consider this problem from the point of view of a one-to-one correspondence between a non-finite unitary operator and an infinite (five-diagonal) block three-diagonal Jacobi-type matrix in the form of the corresponding direct and inverse spectral problems for the trigonometric moment problem. Since the earlier obtained block three-diagonal Jacobi-type unitary matrix has not been fully described, we continue this investigations in the present article. In particular, we show that this exact inner structure coincides with an earlier obtained CMV-matrix.

### The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices

Yurij M. Berezansky, Mykola E. Dudkin

Methods Funct. Anal. Topology **12** (2006), no. 1, 1-31

We continue to generalize the connection between the classical power moment problem and the spectral theory of selfadjoint Jacobi matrices. In this article we propose an analog of the Jacobi matrix related to the complex moment problem and to a system of polynomials orthogonal with respect to some probability measure on the complex plane. Such a matrix has a block three-diagonal structure and gives rise to a normal operator acting on a space of l2 type. Using this connection we prove existence of a one-to-one correspondence between probability measures defined on the complex plane and block three-diagonal Jacobi type normal matrices. For simplicity, we investigate in this article only bounded normal operators. From the point of view of the complex moment problem, this restriction means that the measure in the moment representation (or the measure, connected with the orthonormal polynomials) has compact support.

### The direct and inverse spectral problems for the block Jacobi type unitary matrices

Yurij M. Berezansky, Mykola E. Dudkin

Methods Funct. Anal. Topology **11** (2005), no. 4, 327-345

### Dense subspaces in scales of Hilbert spaces

S. Albeverio, R. Bozhok, M. Dudkin, V. Koshmanenko

Methods Funct. Anal. Topology **11** (2005), no. 2, 156-169

### The complex moment problem in the exponential form

Yu. M. Berezansky, M. E. Dudkin

Methods Funct. Anal. Topology **10** (2004), no. 4, 1-10