Refinement of von Neumann-type inequalities on product Eaton triples

In this paper, a von Neumann-type inequality is studied on an Eaton triple $ (V,G,D) $, where $ V $ is a real inner product space, $ G $ is a compact subgroup of the orthogonal group $ O (V) $, and $ D \subset V $ is a closed convex cone. By using an inner structure of an Eaton triple, a refinement of this inequality is shown. In the special case $ G = O ( V ) $, a refinement of the Cauchy-Schwarz inequality is obtained.