arXiv subjects
Publications and source records attributed to T. Constantinescu.
We discuss the structure of positive definite kernels in terms of operator models. In particular, we introduce two models, one of Hessenberg type and another one that we call near triangular. These models produce parametrizations of the kernels and we describe the combinatorial nature of these parametrizations in terms of lattice paths of Dyck and Lukasiewicz type.
In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.
In this paper we investigate the class of invariant positive definite kernels on the free semigroup on N generators. We provide a combinatorial description of the positivity of the kernel in terms of Dyck paths and then we find a displacement equation that encodes the invariance property of the kernel.
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on some dilation theoretic techniques that are also described in some details.
Motivated by the remarkable interplay between (chordal) graphs and matrix algebra, we associate to each graph a so-called completion number that might encode some aspects of that interplay. We show that this number is not trivial, and we ask for a graph theoretic characterization of those graphs with a given completion number.
In this manuscript, a parametrization of positive matrices together with some of its many applications in quantum information theory is given.
We continue to investigate some classes of Szegö type polynomials in several variables. We focus on asymptotic properties of these polynomials and we extend several classical results of G. Szegö to this setting.
In previous works we analysed conditions for linearization of hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real space generated by positive definite kernels. The aim of this note is to find more concrete expressions of the Schwartz type conditions: in the Hamburger moment problem for Hankel type kernels on the free semigroup, in dilation theory (Stinespring type dilations and Haagerup decomposability), as well as in multi-variable holomorphy. Among other things, we prove that any hermitian holomorphic kernel has a holomorphic linearization, and hence that holomorphic kernels automatically satisfy L. Schwartz's boundedness condition.
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations for these polynomials, Christoffel-Darboux formulas, and Jacobi type matrices.
In this paper we continue to explore the connection between tensor algebras and displacement structure. We focus on recursive orthonormalization and we develop an analogue of the Szego type theory of orthogonal polynomials in the unit circle for several noncommuting variables. Thus, we obtain the recurrence equations and Christoffel-Darboux type formulas, as well as a Favard type result. Also we continue to study a Szego kernel for the N-dimnesional unit ball of an infinite dimensional Hilbert space.
An indefinite generalization of Nudel'man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides results on existence criteria for Pick-Nevanlinna and Caratheodory-Fejer interpolation, the method yealds new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel'man's problem.
We suggest that a certain one-to-one parametrization of completely positive maps on the matrix algebra might be useful in the study of quantum channels. This is illustrated in the case of binary quantum channels. While the algorithm is quite intricate, it admits a simple, lattice structure representation.
In this paper we explore the connection between tensor algebras and displacement structure. We describe a scattering experiment in this framework, we obtain a realization of the elements of the tensor algebra as transfer maps of a certain class of nonstationary linear systems, and we describe a Schur algorithm for the Schur elements of the tensor algebra.
In this paper we formulate and solve Nevanlinna-Pick and Carathéodory type problems for tensor algebras with data given on the N-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory.
Realization theory for operator colligations on Pontryagin spaces is used to study interpolation and factorization in generalized Schur classes. Several criteria are derived which imply that a given function is almost the restriction of a generalized Schur function. The role of realization theory in coefficient problems is also discussed.