Sharp nonzero lower bounds for the Schur product theorem

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.