On Schatten p-norms of commutators

Abstract

For any odd integer d≥3, we determine the sharpest constant Cp,q,r such that‖XY−YX‖p≤Cp,q,r‖X‖q‖Y‖r for all X,Y∈Md, where Md denotes the set of all d×d complex matrices, ‖⋅‖p, 1≤p≤∞, denotes the Schatten p-norm on Md, and 1≤p,q,r≤∞ satisfy 1p>1q+1r. This is a continuation of the study of the problem considered in Wenzel and Audenaert (2010) [8].