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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sharp nonzero lower bounds for the Schur product theorem
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by Apoorva Khare PDF
Proc. Amer. Math. Soc. 149 (2021), 5049-5063 Request permission


By a result of Schur [J. Reine Angew. Math. 140 (1911), pp. 1–28], the entrywise product $M \circ N$ 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 $M \circ \overline {M} \geq E_n / n$ for all $n \times n$ real or complex correlation matrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle a conjecture of Novak [J. Complexity 15 (1999), pp. 299–316] 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 $M \circ N$ when $N \neq M, \overline {M}$. A natural third question is to ask for a tighter lower bound that does not vanish as $n \to \infty$, i.e., over infinite-dimensional Hilbert spaces.

In this note, we affirmatively answer all three questions by extending and refining Vybíral’s result to lower-bound $M \circ N$, for arbitrary complex positive semidefinite matrices $M,N$. Specifically: we provide tight lower bounds, improving on Vybíral’s bounds. Second, our proof is ‘conceptual’ (and self-contained), providing a natural interpretation of these improved bounds via tracial Cauchy–Schwarz inequalities. Third, we extend our tight lower bounds to Hilbert–Schmidt operators. As an application, we settle Open Problem 1 of Hinrichs–Krieg–Novak–Vybíral [J. Complexity 65 (2021), Paper No. 101544, 20 pp.], which yields improvements in the error bounds in certain tensor product (integration) problems.

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Additional Information
  • Apoorva Khare
  • Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore – 560012, India; and Analysis & Probability Research Group, Bangalore – 560012, India
  • MR Author ID: 750359
  • ORCID: 0000-0002-1577-9171
  • Email:
  • Received by editor(s): November 18, 2020
  • Received by editor(s) in revised form: February 3, 2021, February 4, 2021, and February 19, 2021
  • Published electronically: September 21, 2021
  • Additional Notes: This work was partially supported by Ramanujan Fellowship grant SB/S2/RJN-121/2017, MATRICS grant MTR/2017/000295, and SwarnaJayanti Fellowship grants SB/SJF/2019-20/14 and DST/SJF/MS/2019/3 from SERB and DST (Govt. of India), and by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India)
  • Communicated by: Javad Mashreghi
  • © Copyright 2021 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 5049-5063
  • MSC (2020): Primary 15B48, 47B10; Secondary 15A45, 42A82, 43A35, 46C05, 47A63
  • DOI:
  • MathSciNet review: 4327414