Sharp nonzero lower bounds for the Schur product theorem

By a result of Schur [J. Reine Angew. Math. 1911], the entrywise product $M circ N$ of two positive semidefinite matrices $M,N$ is again positive. Vybiral [Adv. Math. 2020] 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 1999] and to positive definite functions on groups. Vybiral (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 $N eq M, overline{M}$. A natural third question is to obtain a tighter lower bound that need 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 Vybirals result to lower-bound $M circ N$, for arbitrary complex positive semidefinite matrices $M, N$. Specifically: we provide tight lower bounds, improving on Vybirals 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-Vybiral [J. Complexity, in press], which yields improvements in the error bounds in certain tensor product (integration) problems.