arXiv subjects

Yury Arlinskii

Publications and source records attributed to Yury Arlinskii.

14 recordsLinked to original sources

arXiv2025

On $C$-Symmetric and $C$-Self-adjoint Unbounded Operators on Hilbert Space

Let $C$ be a conjugation on a Hilbert space $\mathcal{H}$. A densely defined linear operator $A$ on $\mathcal{H}$ is called $C$-symmetric if $CAC\subseteq A^*$ and $C$-self-adjoint if $CAC=A^*$. Our main results describe all $C$-self-adjoint extensions of $A$ on $\mathcal{H}$. Further, we prove a $C$-self-adjointness criterion based on quasi-analytic vectors and we characterize $C$-self-adjoint operators in terms of their polar decompositions.

math.FA↗

arXiv2025

$C$-self-adjoint contractive extensions of $C$-symmetric non-densely defined contractions

Given a conjugation (involution) $C$ on a Hilbert space, $C$-self-adjoint contractive extensions of a non-densely defined $C$-symmetric contraction are studied and parameterizations of all such extensions are obtained. As an application, a proof of an announced result of Glazman \cite{Glazman1} on the existence of maximal dissipative $C$-self-adjoint extensions of a densely defined $C$-symmetric dissipative operator is given.

math.FA↗

arXiv2024

Squares of symmetric operators

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{{\mathcal T}_z\}_{{\rm Im\,} z>0}$ of closed densely defined symmetric operators with the properties: (I) the domain of ${\mathcal T}_z^2$ is a core of ${\mathcal T}_z$, (II) the domain of ${\mathcal T}_z^2$ is dense but note a core of ${\mathcal T}_z$, (III) the domain of ${\mathcal T}_z^2$ is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case ${\rm dom\,} {\mathcal T}_z^2=\{0\}$ has been considered in [5]. Given a densely defined closed symmetric operator $S$, in terms of the intersection of the domain of $S$ with ${\rm ran\,} (S-λI)$ and the projection of the domain of the adjoint $S^*$ on ${\rm ran\,} (S-λI)$, $λ\in{\mathbb C}\setminus{\mathbb R}$, necessary and sufficient conditions for the cases (I)--(III) related to the domain of $S^2$, are obtained.

math.FA↗

arXiv2020

Everything is possible for the domain intersection dom T \cap dom T*

This paper shows that for the domain intersection $\dom T\cap\dom T^*$ of a closed linear operator and its Hilbert space adjoint everything is possible for very common classes of operators with non-empty resolvent set. Apart from the most striking case of a maximal sectorial operator with $\dom T\cap\dom T^*=\{0\}$, we construct classes of operators for which $\dim(\dom T\cap\dom T^*)= n \in \dN_0$; $\dim(\dom T\cap\dom T^*)= \infty$ and at the same time $\codim(\dom T\cap\dom T^*)=\infty$; and $\codim(\dom T\cap\dom T^*)= n \in \dN_0$; the latter includes~the case that $\dom T\cap\dom T^*$ is dense but no core of $T$ and $T^*$ and the case $\dom T=\dom T^*$ for non-normal $T$. We also show that all these possibilities may occur for operators $T$ with non-empty resolvent set such that either $W(T)=\dC$, $T$ is maximal accretive but not sectorial, or $T$ is even maximal sectorial. Moreover, in all but one subcase $T$ can be chosen with compact resolvent.

math.SP↗

arXiv2015

Compressed resolvents of selfadjoint contractive extensions with exit and holomorphic operator-functions associated with them

Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space ${\mathfrak H}$ with an exit in some larger Hilbert space ${\mathfrak H}\oplus{\mathcal H}$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Kreĭn-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.

math.FA↗

arXiv2013

Around the Van Daele--Schmüdgen theorem

For a {bounded} non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space H and possessing a dense range R we propose a new approach to characterisation of phenomenon concerning the existence of subspaces M\subset H such that M\capR=M^\perp\capR=\{0\}. We show how the existence of such subspaces leads to various {pathological} properties of {unbounded} self-adjoint operators related to von Neumann theorems \cite{Neumann}--\cite{Neumann2}. We revise the von Neumann-Van Daele-Schmüdgen assertions \cite{Neumann}, \cite{Daele}, \cite{schmud} to refine them. We also develop {a new systematic approach, which allows to construct for any {unbounded} densely defined symmetric/self-adjoint operator T infinitely many pairs of its closed densely defined restrictions T_k\subset T such that \dom(T^* T_{k})=\{0\} (\Rightarrow \dom T_{k}^2=\{0\}$) k=1,2 and \dom T_1\cap\dom T_2=\{0\}, \dom T_1\dot+\dom T_2=\dom T.

math.FA↗

arXiv2013

Q-functions and boundary triplets of nonnegative operators

Operator-valued $Q$-functions for special pairs of nonnegative selfadjoint extensions of nonnegative not necessarily densely defined operators are defined and their analytical properties are studied. It is shown that the Kre\uın-Ovcharenko statement announced in \cite{KrO2} is valid only for $Q$-functions of densely defined symmetric operators with finite deficiency indices. A general class of boundary triplets for a densely defined nonnegative operator is constructed such that the corresponding Weyl functions are of Kre\uın-Ovcharenko type.

math.FA↗

arXiv2011

Schur parameters, Toeplitz matrices, and Kre\uın shorted operators

We establish connections between Schur parameters of the Schur class operator-valued functions, the corresponding simple conservative realizations, lower triangular Toeplitz matrices, and Kre\uın shorted operators. By means of Schur parameters or shorted operators for defect operators of Toeplitz matrices necessary and sufficient conditions for a simple conservative discrete-time system to be controllable/observable and for a completely non-unitary contraction to be completely non-isometric/completely non-co-isometric are obtained. For the Schur problem a characterization of central solution and uniqueness criteria to the solution are given in terms of shorted operators for defect operators of contractive Toeplitz matrices, corresponding to data.

math.FA↗

arXiv2008

Numerical Range and Quasi-Sectorial Contractions

We apply a method developed by one of the authors, see \cite{Arl1}, to localize the numerical range of \textit{quasi-sectorial} contractions semigroups. Our main theorem establishes a relation between the numerical range of quasi-sectorial contraction semigroups $\{\exp(- t S)\}_{t\ge 0}$, and the maximal {sectorial} generators $S$. We also give a new prove of the rate $O(1/n)$ for the operator-norm Euler formula approximation: $\exp(- t S)=\lim\limits_{n\to \infty}(I+tS/n)^{-n}$, $t\ge 0$, for this class of semigroups.

math.FA↗

arXiv2008

Iterates of the Schur class operator-valued function and their conservative realizations

Let $\mathfrak M$ and $\mathfrak N$ be separable Hilbert spaces and let $Θ(λ)$ be a function from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$ of contractive functions holomorphic on the unit disk. The operator generalization of the classical Schur algorithm associates with $Θ$ the sequence of contractions (the Schur parameters of $Θ$) $Γ_0=Θ(0)\in \bL(\sM,\sN), Γ_n\in\bL(\sD_{Γ_{n-1}}, \sD_{Γ^*_{n-1}}) $ and the sequence of functions $Θ_0 = Θ$, $Θ_n\in {\bf S}(\sD_{Γ_n},\sD_{Γ^*_n})$ $ n=1,...$ (the Schur iterares of $Θ$) connected by the relations \[ Γ_n=Θ_n(0), Θ_n(λ) = Γ_n+λD_{Γ^*_n} Θ_{n+1}(λ) (I + λΓ^*_nΘ_{n+1} (λ))^{-1}D_{Γ_n}, |λ|<1. \] The function $Θ(λ)\in {\bf S}(\sM,\sN)$ can be realized as the transfer function \[ Θ(λ)=D+λC(I-λA)^{-1}B \] of a linear conservative and simple discrete-time system $τ= {\begin{bmatrix}D & C \cr B & A\end{bmatrix}; \mathfrak M, \mathfrak N,\mathfrak H}$ with the state space $\mathfrak H$ and the input and output spaces $\mathfrak M$ and $\mathfrak N $, respectively. In this paper we give a construction of conservative and simple realizations of the Schur iterates $Θ_n$ by means of the conservative and simple realization of $Θ$.

math.FA↗

arXiv2008

Conservative discrete time-invariant systems and block operator CMV matrices

It is well known that an operator-valued function $Θ$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $Θ$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called "Schur parameters". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are the analogs of Cantero--Moral--Velázquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

math.FA↗

arXiv2007

The Kalman--Yakubovich--Popov inequality for passive discrete time-invariant systems

We consider the Kalman - Yakubovich - Popov (KYP) inequality \[ \begin{pmatrix} X-A^* XA-C^*C & -A^*X B- C^*D\cr -B^*X A-D^* C & I- B^*X B-D^*D \end{pmatrix} \ge 0 \] for contractive operator matrices $ \begin{pmatrix} A&B\cr C &D \end{pmatrix}:\begin{pmatrix}\mathfrak{H}\cr\mathfrak{M} \end{pmatrix}\to\begin{pmatrix}\mathfrak{H}\cr\mathfrak{N} \end{pmatrix}, $ where $\mathfrak{H},$ $\mathfrak{M}$, and $\mathfrak{N}$ are separable Hilbert spaces. We restrict ourselves to the solutions $X$ from the operator interval $[0, I_\mathfrak{H}]$. Several equivalent forms of KYP are obtained. Using the parametrization of the blocks of contractive operator matrices, the Kre\uın shorted operator, and the Möbius representation of the Schur class operator-valued function we find several equivalent forms of the KYP inequality. Properties of solutions are established and it is proved that the minimal solution of the KYP inequality satisfies the corresponding algebraic Riccati equation and can be obtained by the iterative procedure with the special choice of the initial point. In terms of the Kre\uın shorted operators a necessary condition and some sufficient conditions for uniqueness of the solution are established.

math.SP↗

arXiv2006

Contractions with rank one defect operators and truncated CMV matrices

The main issue we address in the present paper are the new models for completely non-unitary contractions with rank one defect operators acting on some Hilbert space of dimension $N\leq\infty$. This model complements nicely the well-known models of Liv$\rm{\check{s}}$ic and Sz.-Nagy--Foias. We show that each such an operator is unitarily equivalent to some truncated CMV matrix obtained from the ``full'' CMV matrix by deleting the first row and the first column, and acting in $\ell^2$ ($\dC^N$). This result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to Cantero, Moral and Velázquez. It is shown that another functional model for contractions with rank one defect operators takes the form of the compression $f(ζ)\to P_\cK (ζf(ζ))$ on the Hilbert space $L^2(\dT,dμ)$ with a probability measure $μ$ onto the subspace $\cK=L^2(\dT,dμ)\ominus \dC$. We develop direct and inverse spectral analysis for finite and semi-infinite truncated CMV matrices. In particular, we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving Schur parameters. The uniqueness theorem for recovered truncated CMV matrix from the given mixed spectral data is established. In this part the paper is closely related to the results of Hochstadt and Gesztesy--Simon obtained for finite self-adjoint Jacobi matrices.

math.SP↗

arXiv2006

Non-self-adjoint Jacobi matrices with rank one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank one imaginary part. It is shown that given a set of $n$ not necessarily distinct non-real numbers in the open upper (lower) half-plane uniquely determines a n x n Jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting multiplicity. An algorithm for reconstruction for such finite Jacobi matrices is presented. A new model complementing the well known Livsic triangular model for bounded linear operators with rank one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix and it follows from the fact that any bounded, prime, non-self-adjoint linear operator with rank one imaginary part acting on some finite-dimensional (resp., separable infinite-dimensional Hilbert space) is unitary equivalent to a finite (resp., semi-infinite) non-self-adjoint Jacobi matrix. This result strengthens the classical Stone theorem established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy--Simon uniqueness theorems for finite Jacobi matrices with non-real eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and non-real. A unique Jacobi matrix, unitarily equivalent to the operator of indefinite integration in the Hilbert space L_2[0,l] is found as well as spectral properties of its perturbations and connections with well known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank one imaginary part.

math.SP↗