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 (2019) 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 then 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 obtain a tighter lower bound that need not vanish, over infinite-dimensional Hilbert spaces as well. In this note, we affirmatively answer all three questions by extending and refining Vybiral's result to lower bound $M \circ N$ for arbitrary positive $M, N$. As a special case, the above bound of $E_n/n$ can be improved to $E_n / {\rm rk}(M)$. In addition, our lower bounds - which we show are tracial Cauchy-Schwartz inequalities - are sharp. This is followed by a variant for Hilbert-Schmidt operators. We end with consequences for positive definite functions on groups, metric spaces, and Hilbert spaces.