Solutions To Some Open Problems In Matrix Analysis

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We solve a problem posed by Audenaert and Kittaneh in 2012 by proving that the Aujla–Bourin subadditivity inequality can be strengthened to an optimal form. We also answer questions posed by Bourin and the second author: (1) we significantly extend both the reciprocal Lie–Trotter theorem of Audenaert and Hiai and Kato’s limit theorem; (2) we determine the best constants in inequalities for sums of contractions and positive block matrices, including the sharp triangle inequality [Formula: see text] for three contractions [Formula: see text]; and (3) we prove a remarkable eigenvalue inequality for the symmetric modulus, thereby determining the best dimension-free lower bound in a question studied by Bourin, Lee, and Zhang.