On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

Abstract

This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759-834, 2009) into the structure of the set H ≥ q,2n of all Hankel nonnegative definite sequences, (s j ) 2n j=0 , of complex q × q matrices and its important subclasses H ≥,e q,2n and H > q,2n of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759-834, 2009) a canonical Hankel parametrization [(C k ) n k=1 , (D k ) n k=0 ] consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (s j ) 2n j=0 of complex q × q matrices. The sequences belonging to each of the classes H ≥ q,2n , H ≥,e q,2n , and H > q,2n were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex anal oper theory 3(4):759-834, 2009; Proposition 2.30).In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization [(C k ) n k=1 , (D k ) n k=0 ] of a sequence (s j ) 2n j=0 ∈ H ≥ q,2n , we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (s j ) 2n j=0 (see Theorem 5.5). The matrices [(C k ) n k=1 , (D k ) n k=0 ] will be expressed in terms of an arbitrary monic Communicated by

Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes

“…It explicits connections between interpolation, Leech's factorization theorem (see [9,10]) and the state space method. Next, the paper On a simultaneous approach to the even and odd truncated matricial Hamburger moment by Bernd Fritzsche, Bernd Kirstein and Conrad Mädler, continues the former investigations of the authors on matricial versions of power moment problems (see [4,5,7] and the papers in the volume [1]). The approach is based on Schur analysis, The main tool consists of an appropriate adaptation of the classical algorithm due to I. Schur and R. Nevanlinna to the moment problems under consideration.…”

Relations between the orthogonal matrix polynomials on [a, b], Dyukarev-Stieltjes parameters, and Schur complements

“…Kovalishina [32], [33], Damanik/Pushnitski/Simon [18] and the references therein. See also [20], [21], [27], [19], [36], [26], [16], [17] and [8].…”

From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem

“…Further investigations of OMP on the real line were made by Kovalishina [29], Aptekarev and Nikishin [1], Dym [20], Durán and coauthors [16][17][18][19], Dette and coauthors [11][12][13][14], Damanik/Pushnitski/Simon [10] and the references therein. See also [25][26][27][28]34].…”