arXiv subjects
Publications and source records attributed to Vladimir Bolotnikov.
\We study linear preserver problems on the linear space of $n\times n$ Toeplitz matrices over the real field or the complex field. In particular, characterizations are given for linear preservers of rank one matrices and linear preservers of the determinant. We also present related results and questions on other structured matrices.
We study a general metric constrained interpolation problem in a de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with a contractive multiplier $S$ between two Fock spaces along with its commutative counterpart, a de Branges-Rovnyak space associated with a Schur multiplier on the Drury-Arveson space of the unit ball of $\mathbb{C}^n$.
As is well known, any complex cyclic matrix $A$ is similar to the unique companion matrix associated with the minimal polynomial of $A$. On the other hand, a cyclic matrix over a division ring $\mathbb F$ is similar to a companion matrix of a polynomial which is defined up to polynomial similarity. In this paper we study more rigid canonical forms by embedding a given cyclic matrix over a division ring $\mathbb F$ into a controllable or an observable pair. Using the characterization of ideals in $\mathbb F[z]$ in terms of controllable and observable pairs we consider ideal interpolation schemes in $\mathbb F[z]$ which merge into a polynomial interpolation problems containing both left and right interpolation conditions.
We consider power series over the skew field $\mathbb H$ of real quaternions which are analogous to finite Blaschke products in the classical complex setting. Several intrinsic characteriztions of such series are given in terms of their coefficients as well as in terms of their left and right values. We also discuss the zero structure of finite Blaschke products including left/right zeros and their various multiplicities. We show how to construct a finite Blaschke product with prescribed zero structure. In particular, given a quaternion polynomial $p$ with all zeros less then one in modulus, we explicitly construct a power series $R$ with quaternion coefficients with no zeros such that $pR$ is a finite Blaschke product.
For a division ring $\mathbb F$, the polynomials $f\in\mathbb F$ can be evaluated "on the left" and "on the right" giving rise to left and right Lagrange interpolation problems. The problems containig interpolation conditions of the same type were considered in \cite{lam1} where the solvability criterion was given in terms of polynomial independence of interpolation nodes. We establish the solvability criterion and describe all solutions of low degree (less than the number of interpolation conditions imposed) for the problem containing both "left" and "right" conditions.
It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ which is invariant under the backward shift operator can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the inner Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. The main focus of this paper is to present the extension of this structure to a more general setting. The Hardy space is replaced by the full weighted Bergman-Fock space of formal power series in $d$ freely noncommutative indeterminates, where the shift is replaced by the right shift tuple, where the conservative/dissipative discrete-time linear system becomes a certain type of conservative/dissipative multidimensional linear system with time-varying weights and with evolution along a rooted tree with each node having $d$ forward branches, where a backward shift-invariant subspace ${\mathcal N}$ is the range of the observability operator for such a weighted-Bergman multidimensional linear system, and where the transfer function of this system is the Beurling-Lax representer for the forward shift-invariant subspace ${\mathcal M} = {\mathcal N}^{[\perp]}$, and where this transfer function also serves as the characteristic function for the operator tuple having hypercontractive-operator-tuple adjoint equal to the restriction of the backward-shift tuple to $\mathcal N$.
A general interpolation problem with operator argument is studied for functions f from the de Branges-Rovnyak space H(s) associated with an analytic function s mapping the open unit disk D into the closed unit disk. The interpolation condition is taken in the Rosenblum-Rovnyak form f(A)c = b (with a suitable interpretation of f(A)c) for given Hilbert space operator A and two vectors b; c from the same space.
Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.
We revisit four approaches to the BiTangential Operator Argument Nevanlinna-Pick (BTOA-NP) interpolation theorem on the right half plane: (1) the state-space approach of Ball-Gohberg-Rodman, (2) the Fundamental Matrix Inequality approach of the Potapov school, (3) a reproducing kernel space interpretation for the solution criterion, and (4) the Grassmannian/Kre\uın-space geometry approach of Ball-Helton. These four approaches lead to three distinct solution criteria which therefore must be equivalent to each other. We give alternative concrete direct proofs of each of these latter equivalences. In the final section we show how all the results extend to the case where one seeks to characterize interpolants in the Kre\uın-Langer generalized Schur class $\cS_κ$ of meromorphic matrix functions on the right half plane, with the integer $κ$ as small as possible.
Given $n$ distinct points $t_1,\ldots,t_n$ on the unit circle $\T$ and equally many target values $\f_1,\ldots,\f_n\in\T$, we describe all Blaschke products $f$ of degree at most $n-1$ such that $f(t_i)=\f_i$ for $i=1,\ldots,n$. We also describe the cases where degree $n-1$ is the minimal possible.
It is known that polynomials over quaternions may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial. In this paper, we equivalently describe the zero structure of a polynomial in terms of its left and right spherical divisors as well as in terms of left and right indecomposable divisors. Several algorithms are proposed to find left/right zeros and left/right spherical divisors of a quaternion polynomial, to construct a polynomial with prescribed zero structure and more generally, to construct the least left/right common multiple of given polynomials. Similar questions are briefly discussed in the setting of quaternion formal power series.
We introduce the notion of a confluent Vandermonde matrix with quaternion entries and discuss its connection with Lagrange-Hermite interpolation over quaternions. Further results include the formula for the rank of a confluent Vandermonde matrix, the representation formula for divided differences of quaternion polynomials and their extensions to the formal power series setting.
The Sylvester equation $AX-XB=C$ is considered in the setting of quaternion matrices. Conditions that are necessary and sufficient for the existence of a unique solution are well-known. We study the complementary case where the equation either has infinitely many solutions or does not have solutions at all. Special attention is given to the case where $A$ and $B$ are respectively, lower and upper triangular two-diagonal matrices (in particular, if $A$ and $B$ are Jordan blocks)
It is well known that a non-constant complex-valued function $f$ defined on the open unit disk $\mathbb D$ is an analytic self-mapping of $\D$ if and only if Pick matrices $\left[ (1-f(z_i)\overline{f(z_j)})/(1-z_i\overline{z}_j)\right]_{i,j=1}^n$ are positive semidefinite for all choices of finitely many points $z_i\in\D$. A stronger version of the "if" part was established by Alan Hindmarsh: if all $3\times 3$ Pick matrices are positive semidefinite, then $f$ is an analytic self-mapping of $\mathbb D$. In this paper, we extend this result to the non-commutative setting of power series over quaternions.
Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring ${\mathbb H}[z]$. The problem containing both "left" and "right" interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in ${\mathbb H}[z]$.
The Sz.-Nagy--Foias model theory for $C_{\cdot 0}$ contraction operators combined with the Beurling-Lax theorem establishes a correspondence between any two of four kinds of objects: shift-invariant subspaces, operator-valued inner functions, conservative discrete-time input/state/output linear systems, and $C_{\cdot 0}$ Hilbert-space contraction operators. We discuss an analogue of all these ideas in the context of weighted Hardy spaces over the unit disk and an associated class of hypercontraction operators.
For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$ and related extension space ${\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article.
For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$. A companion survey provides equivalent definitions and basic properties of these spaces as well as applications to function theory and operator theory. The present survey brings to the fore more recent applications to a variety of more elaborate function theory problems, including $H^\infty$-norm constrained interpolation, connections with the Potapov method of Fundamental Matrix Inequalities, parametrization for the set of all solutions of an interpolation problem, variants of the Abstract Interpolation Problem of Katsnelson, Kheifets, and Yuditskii, boundary behavior and boundary interpolation in de Branges-Rovnyak spaces themselves, and extensions to multivariable and Kre\uın-space settings.