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Sharp nonzero lower bounds for the schur product theorem

Khare, A (2021) Sharp nonzero lower bounds for the schur product theorem. In: Proceedings of the American Mathematical Society, 149 (12). pp. 5049-5063.

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Official URL: https://doi.org/10.1090/proc/15555


By a result of Schur J. Reine Angew. Math. 140 (1911), pp. 1â��28, the entrywise product M â�¦ N of two positive semidefinite matrices M, N is again positive. Vybíral Adv. Math. 368 (2020), p. 9 improved on this by showing the uniform lower bound M â�¦ MÍ� â�¥ En/n for all n Ã� n real or complex correlation matrices M, where En is the all-ones matrix. This was applied to settle a conjecture of Novak J. Complexity 15 (1999), pp. 299â��316 and to positive definite functions on groups. Vybíral (in his original preprint) asked if one can obtain similar uniform lower bounds for higher entrywise powers of M, or for M â�¦ N when N â� M, MÍ�. A natural third question is to ask for a tighter lower bound that does not vanish as n â�� â��, i.e., over infinite-dimensional Hilbert spaces. In this note, we affirmatively answer all three questions by extending and refining Vybíralâ��s result to lower-bound M â�¦ N, for arbitrary complex positive semidefinite matrices M, N. Specifically: we provide tight lower bounds, improving on Vybíralâ��s bounds. Second, our proof is â��conceptualâ�� (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchyâ��Schwarz inequalities. Third, we extend our tight lower bounds to Hilbertâ��Schmidt operators. As an application, we settle Open Problem 1 of Hinrichsâ��Kriegâ��Novakâ��Vybíral J. Complexity 65 (2021), Paper No. 101544, 20 pp., which yields improvements in the error bounds in certain tensor product (integration) problems. © 2021 American Mathematical Society

Item Type: Journal Article
Publication: Proceedings of the American Mathematical Society
Publisher: American Mathematical Society
Additional Information: The copyright for this article belongs to American Mathematical Society
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 17 Nov 2021 10:57
Last Modified: 17 Nov 2021 10:57
URI: http://eprints.iisc.ac.in/id/eprint/70508

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