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We study the operator $\mathcal{A}$ of multiplication by an independent variable in a matrix Sobolev space $W^2(M)$. In the cases of finite measures on $[a,b]$ with $(2\times 2)$ and $(3\times 3)$ real continuous matrix weights of full rank it is shown that the operator $\mathcal{A}$ is symmetrizable. Namely, there exist two symmetric operators $\mathcal{B}$ and $\mathcal{C}$ in a larger space such that $\mathcal{A} f = \mathcal{C} \mathcal{B}^{-1} f$, $f\in D(\mathcal{A})$. As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These conditions involve two symmetric operators in an indefinite metric space., Comment: 11 pages

For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in $n$) and generalized eigenvalues of a differential operator (in $n$) are given. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar $L^2_\mu$ spaces are discussed., Comment: 28 pages

In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^\rho }{ n! } x^n {}_2 F_0(-n,\rho;-;-\frac{1}{x})$, depending on a parameter $\rho\in\mathbb{N}$. Differential equations of orders $\rho+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure., Comment: 11 pages

In this paper we study various difference equations related to Jacobi-type pencils. By a Jacobi-type pencil one means the following pencil: $J_5 - \lambda J_3$, 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. The basic set of solutions for the corresponding $4$-th order difference equation is constructed. Spectral properties of the truncated pencil and some special matrix orthogonality relations are investigated. Classical type orthogonal polynomials satisfying a $4$-th order differential equation are constructed., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1706.02391

We study a generalization of a class of orthonormal polynomials on the real axis. These polynomials satisfy the relation $$ \left({J}_5-\uplambda {J}_3\right)\overrightarrow{p}\left(\uplambda \right)=0, $$ where J 3 is a Jacobi matrix, J 5 is a semiinfinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal, $$ \overrightarrow{p}\left(\uplambda \right)={\left({p}_0\left(\uplambda \right),{p}_1\left(\uplambda \right),{p}_2\left(\uplambda \right),\dots \right)}^T $$ , and the superscript T denotes the operation of transposition, with the initial conditions p 0(λ) = 1 and p 1(λ) = αλ + β, α > 0, β ∈ ℝ.We establish certain orthonormality conditions for the polynomials $$ {\left\{{p}_n\left(\uplambda \right)\right\}}_{n=0}^{\infty } $$ . An explicit example of these polynomials is constructed.

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + \lambda_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained., Comment: 18 pages

We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. A way to construct atomic solutions of the moment problem is indicated., 24 pages

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 $H$)., Comment: 9 pages

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., Comment: 16 pages

We study the truncated two-dimensional moment problem (with rectangular data): to find a non-negative measure $\mu(\delta)$, $\delta\in\mathfrak{B}(\mathbb{R}^2)$, such that $\int_{\mathbb{R}^2} x_1^m x_2^n d\mu = s_{m,n}$, $0\leq m\leq M,\quad 0\leq n\leq N$, 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., Comment: 14 pages

In this paper we study the inverse spectral problem for Jacobi-type pencils. By a Jacobi-type pencil we mean the following pencil $J_5 - \lambda J_3$, 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.

The question of the density of polynomials in some spaces L 2(M) is studied. The following two variants of the measure M and the polynomials are considered: (1) an N × N matrix-valued nonnegative Borel measure on ℝ and vector-valued polynomials p(x) = (p 0(x), p 1(x), …, p N−1(x)), where the p j (x) are complex polynomials and N ∈ ℕ (2) a scalar nonnegative Borel measure on the strip Π = {(x, φ): x ∈ ℝ, ϕ ∈ [−π, π)}, and power-trigonometric polynomials $p(x,\phi ) = \sum\limits_{m = 0}^\infty {\sum\limits_{n = - \infty }^\infty {\alpha _{m,n} x^m e^{in\phi } } } ,\alpha _{m,n} \in \mathbb{C}$ , where only finitely many α m,n are nonzero. We show that the polynomials are dense in L 2(M) if and only if M is the canonical solution of the corresponding moment problem. It should be stressed that we do not impose any additional constraints on the measure, except the existence of moments. Using the known descriptions of the canonical solutions,, we obtain conditions on the density of polynomials in L 2(M). Simultaneously, we establish a model for commuting self-adjoint and unitary operators with spectrum of finite multiplicity.

The paper continues the investigation of the truncated matrix trigonometric moment problem originated by the author in Ukr. Mat. Zh., 63, No. 6, 786–797 (2011). The Nevanlinna formula for the indicated moment problem is deduced in the general case. It is assumed that there is more than one moment and that the moment problem is solvable and has more than one solution. The coefficients of the corresponding matrix linear fractional transformation are expressed in the explicit form via the given moments. Simple determinacy conditions are presented for the moment problem.

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., Comment: Expository paper

We study the truncated matrix trigonometric problem of moments. A parametrization of all solutions of this problem (both in the nondegenerate and degenerate cases) is obtained by using the operator approach. This parametrization establishes the one-to-one correspondence between a certain class of analytic functions and all solutions of the problem. We use the important Chumakin results on the generalized resolvents of isometric operators.

In this paper, we study the necessary and sufficient conditions for the density of the linear space of matrix polynomials in a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays of the complex plane. The connection with a completely indeterminate Hamburger matrix moment problem is stated. Vector valued functions associated in a natural way with a function defined in the union of the radial rays are used. Thus, our first aim is the construction of a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays and a positive semi-definite matrix acting on the discrete part of the corresponding inner product. An isometric transformation which allows to reduce the problem of density to the case of the real line is introduced. Finally, some examples of such spaces are shown and its completeness is studied in detail.

We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators.

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., Comment: 16 pages

16 pages, no figures.-- MSC2000 codes: 42C05. MR#: MR2205410 (2007m:39002) Zbl#: Zbl 1100.42022 In this paper, we analyze the set of solutions of a linear difference equation of high-order. More precisely, we obtain a basis of such a linear subspace. We deduce integral representations for elements of this basis. The research of the first author has been supported by Dirección General de Investigación (Ministerio de Educación y Ciencia) of Spain under grant BFM 2003-06335-C03-02, and INTAS Research Network NeCCA INTAS 03-51-6637. Publicado

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 $M_{\mathbb{T}}(\delta)$, $\delta\in \mathfrak{B}(\mathbb{T})$, generated by the seeked function $M(x)$: $M_{\mathbb{T}}(\Delta) = 0$, where $\Delta$ 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., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1401.5413

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., 5 pages

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., 13 pages

In this paper we study a possibility of a decomposition of a bounded operator in a Hilbert space H as a product of a J-unitary and a J-self-adjoint operators, where J is a conjugation (an antilinear involution). This decompo- sition shows an inner structure of a bounded operator in a Hilbert space. Some decompositions of J-unitary and unitary operators which generalize decompo- sitions in the finite-dimensional case are also obtained. Matrix representations for J-symmetric and J-skew-symmetric operators are studied. Simple basic properties of J-symmetric, J-skew-symmetric and J-isometric operators are obtained.