HAO-CHUNG CHENGMin-Hsiu Hsieh2022-12-142022-12-1420161364-5021https://scholars.lib.ntu.edu.tw/handle/123456789/62641621 pages. Text partially overlaps with arXiv:1506.06801. Accepted in Proceedings of the Royal Society A: Mathematical, Physical & Engineering SciencesWe derive new characterizations of the matrix Φ-entropy functionals introduced in Chen & Tropp (Chen, Tropp 2014 Electron. J. Prob.19, 1-30. (doi:10.1214/ejp.v19-2964)). These characterizations help us to better understand the properties of matrix Φ-entropies, and are a powerful tool for establishing matrix concentration inequalities for random matrices. Then, we propose an operator-valued generalization of matrix Φ-entropy functionals, and prove the subadditivity under Löwner partial ordering. Our results demonstrate that the subadditivity of operator-valued Φ-entropies is equivalent to the convexity. As an application, we derive the operator Efron-Stein inequality.enEfron–Stein inequality; matrix concentration inequalities; Φ-entropy; Mathematical Physics; Mathematical Physics; Computer Science - Information Theory; Mathematics - Information Theory; Mathematics - Mathematical Physics; Mathematics - Probability; Quantum Physicsjournal article10.1098/rspa.2015.0563271189092-s2.0-84962809041WOS:000377720500005http://arxiv.org/abs/1602.00233v151210953