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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 correlationmatrices $M$, where $E_n$ is the all-ones matrix. This was applied to settle aconjecture of Novak [J. Complexity 1999] and to positive definite functions ongroups. Vybiral (in his original preprint) asked if one can obtain similaruniform 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 obtain a tighterlower bound that need not vanish as $n \to \infty$, i.e. overinfinite-dimensional Hilbert spaces.  In this note, we affirmatively answer all three questions by extending andrefining Vybiral's result to lower-bound $M \circ N$, for arbitrary complexpositive semidefinite matrices $M, N$. Specifically: we provide tight lowerbounds, improving on Vybiral's bounds. Second, our proof is 'conceptual' (andself-contained), providing a natural interpretation of these improved boundsvia tracial Cauchy-Schwarz inequalities. Third, we extend our tight lowerbounds to Hilbert-Schmidt operators. As an application, we settle Open Problem1 of Hinrichs-Krieg-Novak-Vybiral [J. Complexity, in press], which yieldsimprovements in the error bounds in certain tensor product (integration)problems.

Sharp nonzero lower bounds for the Schur product theorem