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Victor Vinnikov

Publications and source records attributed to Victor Vinnikov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Null- and Positivstellensätze for rationally resolvable ideals

Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[x]. In the free algebra C the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in finite-dimensional representations of C /I. In this article Nullstellensätze for a simple but important class of ideals in the free algebra - called tentatively rationally resolvable here - are presented. An ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the X variables using noncommutative rational functions in the remaining variables. Whether such an ideal I satisfies the Nullstellensatz is intimately related to embeddability of C /I into (free) skew fields. These notions are also extended to free algebras with involution. For instance, it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two-sided ideal I generated by 1-\sum_j X_j^* X_j. This is then applied to free real algebraic geometry: polynomials positive semidefinite on spherical isometries are sums of Hermitian squares modulo I. Similar results are obtained for nc unitary groups.

math.RA↗
arXiv2016

Representations of Lie Algebras by non-Skewselfadjoint Operators in Hilbert Space

We study non-selfadjoint representations of a finite dimensional real Lie algebra $\fg$. To this end we embed a non-selfadjoint representation of $\fg$ into a more complicated structure, that we call a $\fg$-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group $\fG$. We develop the frequency domain theory of the system in terms of representations of $\fG$, and introduce the joint characteristic function of a $\fg$-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the $ax+b$ group, the group of affine transformations of the line.

math.DS↗
arXiv2016

Matrix-valued Hermitian Positivstellensatz, lurking contractions, and contractive determinantal representations of stable polynomials

We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization $$F(z)= D + C\mathbf{P}(z)_n(I-A\mathbf{P}(z)_n)^{-1} B. $$ Here $\mathcal{D}_\mathbf{P}$ is defined by the inequality $\|\mathbf{P}(z)\|<1$, where $\mathbf{P}(z)$ is a direct sum of matrix polynomials $\mathbf{P}_i(z)$ (so that appropriate Archimedean and approximation conditions are satisfied), and $\mathbf{P}(z)_n=\bigoplus_{i=1}^k\mathbf{P}_i(z)\otimes I_{n_i}$, with some $k$-tuple $n$ of multiplicities $n_i$; special cases include the open unit polydisk and the classical Cartan domains. The proof uses a matrix-valued version of a Hermitian Positivstellensatz by Putinar, and a lurking contraction argument. As a consequence, we show that every polynomial with no zeros on the closure of $\mathcal{D}_\mathbf{P}$ is a factor of $\det (I - K\mathbf{P}(z)_n)$, with a contractive matrix $K$.

math.CV↗
arXiv2016

Dilations of Semigroups of Contractions through Vessels

Let $A_1,\ldots,A_d$ be a $d$-tuple of commuting dissipative operators on a separable Hilbert space $\mathcal{H}$. Using the theory of operator vessels and their associated systems, we give a construction of a dilation of the multi-parameter semigroup of contractions on $\mathcal{H}$ given by $e^{i \sum_{j=1}^d t_j A_j}$.

math.FA↗
arXiv2016

On the multiplication of operator-valued c-free random variables

We discuss some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $( Φ, φ) $, where $ Φ$ is a linear map with values in some Banach or C$^\ast$-algebra and $ φ$ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu's $ S $-transform for this framework.

math.OA↗
arXiv2016

Noncommutative reproducing kernel Hilbert spaces

The theory of positive kernels and associated reproducing kernel Hilbert spaces, especially in the setting of holomorphic functions, has been an important tool for the last several decades in a number of areas of complex analysis and operator theory. An interesting generalization of holomorphic functions, namely free noncommutative functions (e.g., functions of square-matrix arguments of arbitrary size satisfying additional natural compatibility conditions), is now an active area of research, with motivation and applications from a variety of areas (e.g., noncommutative functional calculus, free probability, and optimization theory in linear systems engineering). The purpose of this article is to develop a theory of positive kernels and associated reproducing kernel Hilbert spaces for the setting of free noncommutative function theory.

math.OA↗
arXiv2016

Interpolation and transfer-function realization for the noncommutative Schur-Agler class

The Schur-Agler class consists of functions over a domain satisfying an appropriate von Neumann inequality. Originally defined over the polydisk, the idea has been extended to general domains in multivariable complex Euclidean space with matrix polynomial defining function as well as to certain multivariable noncommutative-operator domains with a noncommutative linear-pencil defining function. Still more recently there has emerged a free noncommutative function theory (functions of noncommuting matrix variables respecting direct sums and similaritytransformations). The purpose of the present paper is to extend the Schur-Agler-class theory to the free noncommutative functionsetting. This includes the positive-kernel-decomposition characterization of the class, transfer-function realization and Pick interpolation theory. A special class of defining functions is identified for which the associated Schur-Agler class coincides with the contractive-multiplier class on an associated noncommutative reproducing kernel Hilbert space; in this case, solution of the Pick interpolation problem is in terms of the complete positivity of an associated Pick matrix which is explicitly determined from the interpolation data.

math.FA↗
arXiv2015

Meromorphic Matrix Trivializations of Factors of Automorphy over a Riemann Surface

It is a consequence of the Jacobi Inversion Theorem that a line bundle over a Riemann surface M of genus g has a meromorphic section having at most g poles, or equivalently, the divisor class of a divisor D over M contains a divisor having at most g poles (counting multiplicities). We explore various analogues of these ideas for vector bundles and associated matrix divisors over M. The most explicit results are for the genus 1 case. We also review and improve earlier results concerning the construction of automorphic or relatively automorphic meromorphic matrix functions having a prescribed null/pole structure.

math.CV↗
arXiv2015

Livsic-type Determinantal Representations and Hyperbolicity

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety $X \subset \mathbb{P}^d$ of an arbitrary codimension $\ell$ with respect to a real $\ell - 1$-dimensional linear subspace $V \subset \mathbb{P}^d$ and study its basic properties. We also consider a special kind of determinantal representations that we call Livsic-type and a nice subclass of these that we call \vr{}. Much like in the case of hypersurfaces ($\ell=1$), the existence of a definite Hermitian \vr{} Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a \vr{} Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in $\mathbb{P}^d$ hyperbolic with respect to some real $d-2$-dimensional linear subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type determinantal representation.

math.AG↗
arXiv2015

Rational inner functions on a square-matrix polyball

We establish the existence of a finite-dimensional unitary realization for every matrix-valued rational inner function from the Schur--Agler class on a unit square-matrix polyball. In the scalar-valued case, we characterize the denominators of these functions. We also show that every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Korányi--Vagi theorem generalizing Rudin's description of rational inner functions to the case of bounded symmetric domains; we provide a short elementary proof of this theorem suitable in our setting.

math.CV↗
arXiv2015

Contractive determinantal representations of stable polynomials on a matrix polyball

We show that an irreducible polynomial $p$ with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that $p(0)=1$, admits a strictly contractive determinantal representation, i.e., $p=\det(I-KZ_n)$, where $n=(n_1,...,n_k)$ is a $k$-tuple of nonnegative integers, $Z_n=\bigoplus_{r=1}^k(Z^{(r)}\otimes I_{n_r})$, $Z^{(r)}=[z^{(r)}_{ij}]$ are complex matrices, $p$ is a polynomial in the matrix entries $z^{(r)}_{ij}$, and $K$ is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.

math.FA↗
arXiv2015

Convex entire noncommutative functions are polynomials of degree two or less

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result: that a function of $x$ that is matrix convex near $0$ and also that is "analytic" in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function $F$ in two classes of noncommuting variables, $a = (a_1, \ldots, a_{\tilde{g}})$ and $x = (x_1, \ldots, x_g)$ that is "analytic" and matrix convex in $x$ on a "noncommutative open set" in $a$ is a polynomial of degree two or less.

math.FA↗
arXiv2013

Scattering systems with several evolutions and formal reproducing kernel Hilbert spaces

A Schur-class function in $d$ variables is defined to be an analytic contractive-operator valued function on the unit polydisk. Such a function is said to be in the Schur--Agler class if it is contractive when evaluated on any commutative $d$-tuple of strict contractions on a Hilbert space. It is known that the Schur--Agler class is a strictly proper subclass of the Schur class if the number of variables $d$ is more than two. The Schur--Agler class is also characterized as those functions arising as the transfer function of a certain type (Givone--Roesser) of conservative multidimensional linear system. Previous work of the authors identified the Schur--Agler class as those Schur-class functions which arise as the scattering matrix for a certain type of (not necessarily minimal) Lax--Phillips multievolution scattering system having some additional geometric structure. The present paper links this additional geometric scattering structure directly with a known reproducing-kernel characterization of the Schur--Agler class. We use extensively the technique of formal reproducing kernel Hilbert spaces that was previously introduced by the authors and that allows us to manipulate formal power series in several commuting variables and their inverses (e.g., Fourier series of elements of $L^2$ on a torus) in the same way as one manipulates analytic functions in the usual setting of reproducing kernel Hilbert spaces.

math.FA↗
arXiv2013

Stable and real-zero polynomials in two variables

For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an $(n_1+n_2)\times(n_1+n_2)$ diagonal matrix with coordinate variables $z_1$, $z_2$ on the diagonal and $K$ is a contraction. We show that $K$ may be chosen to be unitary if and only if $p$ is a (unimodular) constant multiple of its reverse. Furthermore, for every bivariate real-zero polynomial $p(x_1, x_2),$ with $p(0,0)=1$, we provide a construction to build a representation of the form $$p(x_1,x_2)=\det (I+x_1A_1+x_2A_2),$$ where $A_1$ and $A_2$ are Hermitian matrices of size equal to the degree of $p$. A key component of both constructions is a stable factorization of a positive semidefinite matrix-valued polynomial in one variable, either on the circle (trigonometric polynomial) or on the real line (algebraic polynomial).

math.FA↗
arXiv2012

Determinantal representations of singular hypersurfaces in P^n

A (global) determinantal representation of hypersurface in P^n is a matrix, whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface. We study the properties of such representations for singular (possibly reducible or non-reduced) hypersurfaces. In particular, we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces. Further, we classify the determinantal representations in terms of the corresponding kernel sheaves on $X$. Finally, we extend the results to the case of symmetric/self-adjoint representations, with implications to hyperbolic polynomials and generalized Lax conjecture.

math.AG↗
arXiv2011

Decomposability of local determinantal representations of hypersurfaces

Let M be a matrix whose entries are power series in several variables and determinant det(M) does not vanish identically. The equation det(M)=0 defines a hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay module over the local ring of this singularity. Suppose the determinant det(M) is reducible, i.e. the hypersurface is locally reducible. A natural question is whether the matrix is equivalent to a block-diagonal or at least to an upper-block-triangular. (Or whether the corresponding module is decomposable or at least is an extension.) We give various necessary and sufficient criteria. Two classes of such matrices of functions appear naturally in the study of decomposability: those with many generators (e.g. maximally generated or Ulrich maximal) and those that descend from birational modifications of the hypersurface by pushforwards (i.e. correspond to modules over bigger rings). Their properties are studied.

math.AG↗