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By a result of Schur [J. Reine Angew. Math. 140 (1911), pp. 1–28], the entrywise product M ∘ 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 ∘ M ≥ En/n for all n × n real or complex correlation matrices M, where En 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 ∘ N when N ≠ M, M. A natural third question is to ask for a tighter lower bound that does not vanish as n → ∞, 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 ∘ 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. [ABSTRACT FROM AUTHOR]

We provide an elementary proof of a result by V. P. Fonf and C. Zanco on point-finite coverings of separable Hilbert spaces. Indeed, by using a variation of the famous argument introduced by J. Lindenstrauss and R. R. Phelps [Israel J. Math. 6 (1968), pp. 39–48] to prove that the unit ball of a reflexive infinite-dimensional Banach space has uncountably many extreme points, we prove the following result. Let X be an infinite-dimensional Hilbert space satisfying dens(X)< 2ℵ0, then X does not admit point-finite coverings by open or closed balls, each of positive radius. In the second part of the paper, we follow the argument introduced by V. P. Fonf, M. Levin, and C. Zanco in [J. Geom. Anal. 24 (2014), pp. 1891–1897] to prove that the previous result holds also in infinite-dimensional Banach spaces that are both uniformly rotund and uniformly smooth. [ABSTRACT FROM AUTHOR]