Journal of the
Korean Mathematical Society JKMS

A matrix completion problem has been exploited amply because of its abundant applications and the analysis of contractions enables us to have insight into structure and space of operators. In this article, we focus on a specific completion problem related to Hankel partial contractions. 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 \cite{CLY}. 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\times n $ in both types.

Keywords: Hankel partial contraction, contractive completion, extremal type, non-extremal type

MSC numbers: Primary 47B35, 15A83, 15A60, 15A48; Secondary 47A30, 15-04