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

## Abstract

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. VybÃral 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. VybÃral (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 VybÃralâ��s result to lower-bound M â�¦ N, for arbitrary complex positive semidefinite matrices M, N. Specifically: we provide tight lower bounds, improving on VybÃralâ��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��VybÃral J. Complexity 65 (2021), Paper No. 101544, 20 pp., which yields improvements in the error bounds in certain tensor product (integration) problems. Â© 2021 American Mathematical Society

Item Type: | Journal Article |
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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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