Schur products and matrix completions

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Abstract

We prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain Schur product map defined on a certain subspace of matrices is a positive map. By analyzing the positive elements of this subspace, we obtain new proofs of the results of H. Dym and I. Gohberg and Grone, Johnson, Sa, and Wolkowitz (Linear Algebra Appl.58 (1984), 109-124). (Linear Algebra Appl.36 (1981), 1-24). We also obtain a new proof of a result of U. Haagerup (Decomposition of completely bounded maps on operation algebras, preprint), characterizing the norm of Schur product maps, and a new Hahn-Banach type extension theorem for these maps. Finally, we obtain generalizations of many of these results to matrices of operators, which we apply to the study of representations of certain subalgebras of the n × n matrices. © 1989.

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Paulsen, V. I., Power, S. C., & Smith, R. R. (1989). Schur products and matrix completions. Journal of Functional Analysis, 85(1), 151–178. https://doi.org/10.1016/0022-1236(89)90050-5

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