Dmitry S. Kaliuzhnyi-Verbovetskyi

Noncommutative rational functions, their difference-differential calculus and realizations

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.

Conservative dilations of dissipative multidimensional systems: The commutative and non-commutative settings

We establish the existence of conservative dilations for various types of dissipative non-commutative N-dimensional (N-D) systems. As a corollary, a criterion of existence of conservative dilations for corresponding dissipative commutative N-D systems is obtained. We point out the cases where this criterion is always fulfilled, and the cases where it is not always fulfilled.

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 \(\mathcal{D}_{p}\; \mathrm{in}\;\mathbb{C}^{d}\) and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization

Noncommutative formal power series and noncommutative functions

F(z)\;=\;D\;+\;CP(z)_{n}(I-AP(z)_n)^{-1}B

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

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Singularities of rational functions and minimal factorizations : The noncommutative and the commutative setting

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 a multiple of every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Koranyi–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.

Stable and real-zero polynomials in two variables

In various applications of formal power series, their evaluations on linear operators (acting on an infinite-dimensional Hilbert space) or on square matrices (of any size or of size large enough) play an important role and allow one to develop a noncommutative analog of analytic function theory. On the other hand, functions defined on square matrices of any size which respect direct sums and similarities and satisfy a local boundedness condition behave in many ways as analytic functions and have power series expansions — a noncommutative analogue of Taylor series. We will discuss convergence of noncommutative power series and analyticity of noncommutative functions.