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 ? N M \circ N two positive semidefinite matrices comma , encoding="application/x-tex">M,N is again positive. Vybíral [Adv. 368 (2020), p. 9] improved on this by showing uniform lower bound overbar greater-than-or-equal-to E Subscript n Baseline slash n"> ¯ \to \infty i.e., over infinite-dimensional Hilbert spaces. In note, we affirmatively answer three questions extending refining Vybíral’s lower-bound arbitrary Specifically: provide tight bounds, improving bounds. Second, our proof ‘conceptual’ (and self-contained), providing interpretation these via tracial Cauchy–Schwarz inequalities. Third, extend Hilbert–Schmidt operators. As an application, Open Problem 1 Hinrichs–Krieg–Novak–Vybíral 65 (2021), Paper No. 101544, 20 pp.], which yields improvements in error certain tensor (integration) problems.