arXiv subjects
Publications and source records attributed to Sergey M. Zagorodnyuk.
In this paper we study the inverse spectral problem for Jacobi-type pencils. By a Jacobi-type pencil we mean the following pencil $J_5 - λJ_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. In the case of a special perturbation of orthogonal polynomials on a finite interval the corresponding spectral function takes an explicit form.
We study the truncated two-dimensional moment problem (with rectangular data): to find a non-negative measure $μ(δ)$, $δ\in\mathfrak{B}(\mathbb{R}^2)$, such that $\int_{\mathbb{R}^2} x_1^m x_2^n dμ= s_{m,n}$, $0\leq m\leq M,\quad 0\leq n\leq N$, where $\{ s_{m,n} \}_{0\leq m\leq M,\ 0\leq n\leq N}$ is a prescribed sequence of real numbers; $M,N\in\mathbb{Z}_+$. For the cases $M=N=1$ and $M=1, N=2$ explicit numerical necessary and sufficient conditions for the solvability of the moment problem are given. In the cases $M=N=2$; $M=2, N=3$; $M=3, N=2$; $M=3, N=3$ some explicit numerical sufficient conditions for the solvability are obtained. In all the cases some solutions (not necessarily atomic) of the moment problem can be constructed.
In this paper we obtain a Nevanlinna-type parametrization for the operator Hamburger moment problem. The moment problem is not supposed to be necessarily completely indeterminate. No assumptions besides the solvability of the moment problem are posed.
In this paper we study a generalization of the class of orthogonal polynomials on the real line. These polynomials satisfy the following relation: $(J_5 - λJ_3) \vec p(λ) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $\vec p(λ) = (p_0(λ), p_1(λ), p_2(λ),\cdots)^T$, the superscript $T$ means the transposition, with the initial conditions $p_0(λ) = 1$, $p_1(λ) = αλ+ β$, $α> 0$, $β\in\mathbb{R}$. Some orthonormality conditions for the polynomials $\{ p_n(λ) \}_{n=0}^\infty$ are obtained. An explicit example of such polynomials is constructed.
In this paper we consider a notion of a generalized resolvent for a pair of commuting isometric operators in a Hilbert space $H$. Characteristic properties of the generalized resolvent are obtained.
This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, \textbf{63}, no. 6, 786-797, and Ukrainian Math. J., 2013, \textbf{64}, no. 8, 1199-1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure $M_{\mathbb{T}}(δ)$, $δ\in \mathfrak{B}(\mathbb{T})$, generated by the seeked function $M(x)$: $M_{\mathbb{T}}(Δ) = 0$, where $Δ$ is a given open subset of $\mathbb{T}$ (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.
In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.
In this paper, we shall characterize the components of the polar decomposition for an arbitrary $J$-unitary operator in a Hilbert space. This characterization has a quite different structure as that for complex symmetric and complex skew-symmetric operators. It is also shown that for a $J$-imaginary closed symmetric operator in a Hilbert space there exists a $J$-imaginary self-adjoint extension in a possibly larger Hilbert space (a linear operator $A$ in a Hilbert space $H$ is said to be $J$-imaginary if $f\in D(A)$ implies $Jf\in D(A)$ and $AJf = -JAf$, where $J$ is a conjugation on $H$).
In this paper it is proved that each densely defined $J$-skew-symmetric operator (or each $J$-isometric operator with $\overline{D(A)}=\overline{R(A)}=H$) in a Hilbert space $H$ has a $J$-skew-self-adjoint (respectively $J$-unitary) extension in a Hilbert space $\widetilde H\supseteq H$. We follow the ideas of Galindo in~[A.~Galindo, On the existence of $J$-self-adjoint extensions of $J$-symmetric operators with adjoint, Communications on pure and applied mathematics, Vol. XV, 423-425 (1962)] with necessary modifications.
In this paper we study truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators. Conditions of the solvability are given. Some canonical solutions of the moment problems are constructed. As a by-product, some extension results for $J$-skew-symmetric and $J$-isometric operators are obtained.
In this paper we study the truncated power moment problem with an odd number of prescribed moments. A Nevanlinna-type formula is derived for this moment problem in the case when the moment problem has more than one solution (the indeterminate case). The coefficients of the corresponding linear fractional transformation are expressed explicitly in terms of the given moments. In the determinate case the solution to the moment problem is constructed, as well. Those solutions of the moment problem, which satisfy the following condition: $M(Δ)=0$, where $Δ$ is a prescribed open subset of $\mathbb{R}$, and $M$ is the matrix measure generated by a solution of the moment problem, are described.
In this paper we study invertible extensions of a symmetric operator in a Hilbert space $H$. All such extensions are characterized by a parameter in the generalized Neumann's formulas. Generalized resolvents, which are generated by the invertible extensions, are extracted by a boundary condition among all generalized resolvents in the Shtraus formula.
This exposition paper is devoted to the theory of Abram Vilgelmovich Shtraus and his disciples and followers. This theory studies the so-called generalized resolvents of symmetric and isometric operators in a Hilbert space and provides alternative formulas to the well-known Krein-type formulas. We think that it would be convenient to have an exposition of the corresponding theory with full proofs and necessary corrections.
In this paper we obtain some slight correction and generalization of the results of Ryabtseva on the generalized resolvents for isometric operators with a gap in their spectrum. Also, analogs of some McKelvey's results and a short proof of Inin's formula for the generalized resolvents of an isometric operator are obtained.
In this paper we obtain a Nevanlinna-type formula for the matrix Hamburger moment problem in a general case. We only assume that the problem is solvable and has more that one solution. We express the matrix coefficients of the corresponding linear fractional transformation in terms of the prescribed moments. Necessary and sufficient conditions for the determinacy of the moment problem are given.
This paper is a continuation of our previous investigation on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no.6, 786-797. In the present paper we obtain a Nevanlinna-type formula for this moment problem in a general case. We only assume that we have more than one moment, the moment problem is solvable and the problem has more than one solution. The coefficients of the corresponding matrix linear fractional transformation are explicitly expressed by the prescribed moments. Easy conditions for the determinacy of the moment problem are given.
In this paper we study the Nevanlinna-Pick matrix interpolation problem in the Carathéodory class with infinite data (both in the nondegenerate and degenerate cases). We develop the Szökefalvi-Nagy and Korányi operator approach to obtain an analytic description of all solutions of the problem. Simple necessary and sufficient conditions for the determinacy of the problem are given.
In this paper we obtain necessary and sufficient conditions for a linear bounded operator in a Hilbert space $H$ to have a three-diagonal complex symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in $H$. It is shown that a set of all such operators is a proper subset of a set of all complex symmetric operators with a simple spectrum. Similar necessary and sufficient conditions are obtained for a linear bounded operator in $H$ to have a three-diagonal complex skew-symmetric matrix with non-zero elements on the first sub-diagonal in an orthonormal basis in $H$.