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Completion of Hankel partial contractions of non-extremal type. (English) Zbl 1337.47042

Summary: 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 [R. E. Curto et al., J. Math. Phys. 53, No. 12, 123526, 11 p. (2012; Zbl 1287.15010)]. 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.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A83 Matrix completion problems
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15B48 Positive matrices and their generalizations; cones of matrices

Citations:

Zbl 1287.15010
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