## Sharp nonzero lower bounds for the Schur product theorem

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## Abstract:

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: khare@iisc.ac.in
- 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: https://doi.org/10.1090/proc/15555
- MathSciNet review: 4327414