Completion of Hankel partial contractions of extremal type

Abstract

We find concrete necessary and sufficient conditions for the existence of contractive completions of Hankel partial contractions of size 4 × 4 in the extremal case. Along the way we introduce a new approach that allows us to solve, algorithmically, the contractive completion problem for 4 × 4 Hankel matrices. As an application, we obtain a concrete example of a partially contractive 4 × 4 Hankel matrix which does not admit a contractive completion.

Completion of Hankel Partial Contractions of Non-Extremal Type

“…After applying Lemma 2.5, Corollary 2.6, and Theorem 3.2 in [8] to H 33 (x), we get the desired results.…”

“…In this paper, we improve and extend the main results in [8,Section 4] to the non-extremal type for 4 × 4 matrices and extremal type for 5 × 5 matrices, respectively. We also give a negative answer to the conjecture presented in [8,Remark 4.5].…”

“…In this paper, we improve and extend the main results in [8,Section 4] to the non-extremal type for 4 × 4 matrices and extremal type for 5 × 5 matrices, respectively. We also give a negative answer to the conjecture presented in [8,Remark 4.5]. We find concrete necessary and sufficient conditions for the existence of completion of 4 × 4 and 5 × 5 Hankel partial contractions using the Nested Determinants Test (or Choleski's Algorithm), the Moore-Penrose inverse of a matrix, the Schur product techniques of matrices, and the congruence of the positivity for two matrices.…”

“…In [8,Section 4], R. Curto, S. Lee and the third named author of this paper found necessary and sufficient conditions for the existence of contractive completion of HPC's of the extremal type for 4 × 4 matrices. In this paper, we improve and extend the main results in [8,Section 4] to the non-extremal type for 4 × 4 matrices and extremal type for 5 × 5 matrices, respectively.…”

“…We provide concrete necessary and sufficient conditions for the existence of completion of Hankel partial contractions for both extremal and non-extremal types with lower dimensional matrices. Moreover, we give a negative answer for the conjecture presented in [8]. For our results, we use several tools such as the Nested Determinants Test (or Choleski's Algorithm), the Moore-Penrose inverse, the Schur product techniques, and a congruence of two positive semi-definite matrices; all these suggest an algorithmic approach to solve the contractive completion problem for general Hankel matrices of size n × n in both types.…”