Modulus of convexity for operator convex functions

Creators: Kim, Isaac H.

Abstract

Given an operator convex function f(x), we obtain an operator-valued lower bound for cf(x) + (1 − c)f(y) − f(cx + (1 − c)y), c ∈ [0, 1]. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.

Additional Information

© 2014 AIP Publishing LLC. Received 29 April 2014; accepted 3 July 2014; published online 21 July 2014. I would like to thank Andreas Winter and Alexei Kitaev for many helpful discussions which motivated this work. I would also like to thank Jon Tyson, Mary Beth Ruskai, and Fernando Brandão for helpful discussions. I would also like to thank Lin Zhang for pointing out an error in the original manuscript. Lastly, I thank the anonymous referee who suggested to investigate whether the main result holds for operator convex function of finite order. This research was supported in part by NSF under Grant No. PHY-0803371, by ARO Grant No. W911NF-09-1-0442, and DOE Grant No. DE-FG03-92-ER40701. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.

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