arXiv subjects
Publications and source records attributed to Victor Vinnikov.
We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue of the Bercovici-Pata bijection. An important tool is Voiculescu's subordination property for operator-valued free convolution.
The paper is discussing infinite divisibility in the setting of operator-valued boolean, free and, more general, c-free independences. Particularly, using Hilbert bimodules and non-commutative functions techniques, we obtain analogues of the Levy-Hincin integral representation for infinitely divisible real measures.
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial, p, is noncommutative plurisubharmonic if and only if it has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are finite and f_j, k_j, F are all noncommutative analytic. We also present a theory of noncommutative integration for noncommutative polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper by Greene proves that if the noncommutative complex hessian of p takes positive semidefinite values on a "noncommutative open set" then the noncommutative complex hessian takes positive semidefinite values on all matrix tuples. Thus, p has the form above. The proof in the subsequent paper draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions.
We study of the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these measures are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that generally the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.
Noncommutative rational functions appeared in many contexts in system theory and control, from the theory of finite automata and formal languages to robust control and LMIs. We survey the construction of noncommutative rational functions, their realization theory and some of their applications. We also develop a difference-differential calculus as a tool for further analysis.
This work is a direct continuation of the authors work arXiv:0812.3779v1. A special case of conservative overdetermined time invariant 2D systems is developed and studied. Defining transfer function of such a systems we obtain a class CI of inner functions $S(λ,t_2)$, which are identity for $λ=\infty$, satisfy certain regularity assumptions and intertwines solutions of ODEs with a spectral parameter $λ$. Using translation model, we prove that every function in the class CI can be realized as a transfer function of a certain vessel. The highlight of this theory is a generalization Potapov's theorem, which gives a very special formula for such a function in the form of multiplication of Blacke-Potapov products, corresponding to the discrete spectrum of certain system operator $A_1(t_2)$ and of multiplicative integral, corresponding to the continuous spectrum of $A_1(t_2)$. This theorem is proved under a slightly more restrictive assumptions, then the development of the whole theory. Namely, we suppose that the derivative of the transfer function is a continuous function of $t_2$ for almost all $λ$. At the last part zero/pole interpolation problem for matrix functions in CI is considered and a realization theorem of such functions appeared in arXiv:0812.3779v1 (theorem 8.1) is reproved. Hermitian case is also analyzed and the corresponding realization theorem is proved.
In this work we develop a theory of Vessels. This object arises in the study of overdetermined 2D systems invariant in one of the variables, which are usually called time invariant. To each overdetermined time invariant 2D systems there is associated a vessel, which is a collection of system operators satisfying certain relations and vise versa. Such an invariance forces the theory of vessels to resemble a constant (classical) 1D case and as a result many notions are naturally redefined and most theorems are reproved in this setting. The notion of transfer function and its connection to the overdetermined 2D time invariant system (and the corresponding vessel) is one of the topics of this work. It is well known that multiplicative structure of a transfer function of a 1D system is closely connected to the decomposition of the state space into invariant subspaces of the state operator and we generalize this result to a wider class of functions. This class (denoted by $\boldsymbol {\mathcal I}$) arises as a class of transfer functions, which intertwine solutions of ODEs with spectral parameters. At the end we present solution of factorization problems for finite dimensional case.
The work of M. S. Livšic and his collaborators in operator theory associates to a system of commuting nonselfadjoint operators an algebraic curve. Guided by the notion of rational transformation of algebraic curves, we define the notion of a rational transformation of a system of commuting nonselfadjoint operators.
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theory allows to describe explicitly an image of a plane algebraic curve under rational transformation and to determine the number of common zeroes of two polynomials in two variables on a plane algebraic curve.
We show that a formal power series in $2N$ non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on $N$-tuples of matrices of any size obtained from this series by matrix substitution is positive. We present two versions of this result related to different classes of matrix substitutions. In the general case we consider substitutions of jointly nilpotent $N$-tuples of matrices, and thus the question of convergence does not arise. In the ``convergent'' case we consider substitutions of $N$-tuples of matrices from a neighborhood of zero where the series converges. Moreover, in the first case the result can be improved: the positivity of a non-commutative kernel is guaranteed by the positivity of its values on the diagonal, i.e., on pairs of coinciding jointly nilpotent $N$-tuples of matrices. In particular this yields an analogue of a recent result of Helton on non-commutative sums-of-squares representations for the class of hereditary non-commutative polynomials. We show by an example that the improved formulation does not apply in the ``convergent'' case.
This article concerns the question: which subsets of ${\mathbb R}^m$ can be represented with Linear Matrix Inequalities, LMIs? This gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory. Also before having much hope of representing engineering problems as LMIs by automatic methods one needs a good idea of which problems can and cannot be represented by LMIs. Little is currently known about such problems. In this article we give a necessary condition, we call "rigid convexity", which must hold for a set ${\cC} \in {\mathbb R}^m$ in order for ${\cC}$ to have an LMI representation. Rigid convexity is proved to be necessary and sufficient when $m=2$. This settles a question formally stated by Pablo Parrilo and Berndt Sturmfels in [PSprep].
This paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional as well as multiplicity information included in the vector case). This new formalism unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles over an abstract Riemann surface) with an earlier approach of the authors (where the Riemann surface was assumed to be the normalizing Riemann surface for an algebraic curve embedded in ${\bold C}^2$ with determinantal representation, and the vector bundles were assumed to be presented as the kernels of linear matrix pencils). The main tool is a version of the Cauchy kernel appropriate for flat vector bundles over the Riemann surface. Our formula for the interpolating bundle map (in the special case of a single zero and a single pole) can be viewed as a generalization of the Fay trisecant identity from the usual line bundle case to the vector bundle case in terms of Cauchy kernels. In particular we obtain a new proof of the Fay trisecant identity.