arXiv subjects

Volodymyr Derkach

Publications and source records attributed to Volodymyr Derkach.

5 recordsLinked to original sources

arXiv2026

${\sL}$-resolvents and pseudo-spectral functions of symmetric linear relations in Hilbert spaces

Let $A$ be a closed symmetric operator with the deficiency index $(p,p)$, $p<\infty$, acting in a Hilbert space $\sH$ and let $\sL$ be a subspace of $\sH$. The set of $\sL$-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge $\sL(\subset\sH)$ was described by Kre\uın and Saakyan. The Kre\uın--Saakyan theory of $\sL$-resolvent matrices was extended by Shmul'yan and Tsekanovskii to the case of improper gauge $\sL(\not\subset\sH)$ and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Kre\uın--Saakyan theory of $\sL$-resolvent matrices for symmetric linear relations with improper gauges in Hilbert spaces and extend the known formula for the $\sL$-resolvent matrix in terms of boundary operators to this class of relations. Descriptions of spectral and pseudo-spectral functions of symmetric linear relations with improper gauges are given. The results are applied to linear relations generated by a canonical system.

math.SP↗

arXiv2025

L-resolvents of symmetric linear relations in Pontryagin spaces

Let A be a closed symmetric operator with the deficiency index (p,p), $p<\infty$, acting in a Hilbert space H and let L be a subspace of H. The set of L-resolvents of a densely defined symmetric operator in a Hilbert space with a proper gauge L was described by Krein and Saakyan. The Krein--Saakyan theory of L-resolvent matrix was extended by Shmul'yan and Tsekanovskii to the case of improper gauge L and by Langer and Textorius to the case of symmetric linear relations in Hilbert spaces. In the present paper we find connections between the theory of boundary triples and the Krein--Saakyan theory of L-resolvent matrices for symmetric linear relations with improper gauges in Pontryagin spaces. We extend the known formula for the L-resolvent matrix in terms of boundary operators to this class of relations. The results are applied to the minimal linear relation generated by a canonical system.

math.FA↗

arXiv2023

Indefinite Sturm-Liouville operators in polar form

We consider the indefinite Sturm-Liouville differential expression \[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where $\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in I$ and the coefficients $r$ and $w$ are locally summable and such that $r(x)$ and $(\operatorname{sgn} x) w(x)$ are positive a.e. on $I$. With the differential expression $\mathfrak{a}$ we associate a nonnegative self-adjoint operator $A$ in the Krein space $L^2_w(I)$, which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of $I$ with the positive and the negative semi-axis. For the operator $A$ we derive conditions in terms of the coefficients $w$ and $r$ for the existence of a Riesz basis consisting of generalized eigenfunctions of $A$ and for the similarity of $A$ to a self-adjoint operator in a Hilbert space $L^2_{|w|}(I)$. These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces, which are couplings of two symmetric operators acting in Hilbert spaces.

math.SP↗

arXiv2020

On a class of integral systems

We study spectral problems for two--dimensional integral system with two given non-decreasing functions $R_1$, $R_2$ on an interval $[0,b)$ which is a generalization of the Krein string. Associated to this system are the maximal linear relation $T_{\max}$ and the minimal linear relation $T_{\min}$ in the space $L^2(R_2)$ which are connected by $T_{\max}=T_{\min}^*$. It is shown that the limit point condition at $b$ for this system is equivalent to the strong limit point condition for the linear relation $T_{\max}$. In the limit circle case the strong limit point condition fails to hold on $T_{\max}$ but it is still satisfied on a subspace $T_N^*$ of $T_{\max}$ characterized by the Neumann boundary condition at $b$. The notion of the principal Titchmarsh-Weyl coefficient of this integral system is introduced both in the limit point case and in the limit circle case. Boundary triples for the linear relation $T_{\max}$ in the limit point case (and for $T_{N}^*$ in the limit circle case) are constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh-Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of $R_1$ and $R_2$. It is shown that the principal Titchmarsh-Weyl coefficients $q$ and $\widehat q$ of the direct and the dual integral systems are related by the equality $λ\widehat q(λ) = -1/q(λ)$ both in the regular and the singular case.

math.CA↗

arXiv2015

Generalized $γ$-generating matrices and Nehari-Takagi problem

Under certain mild assumption, we establish a one-to-one correspondence between solutions of the Nehari-Takagi problem and solutions of some Takagi-Sarason interpolation problem. The resolvent matrix of the Nehari-Takagi problem is shown to belong to the class of so-called generalized $γ$-generating matrices, which is introduced and studied in the paper.

math.FA↗