Continuous generalization of Clarkson-Mccarthy inequalities

Abstract

Let G be a compact Abelian group, let μ be the corresponding Haar measure, and let Ĝ be the Pontryagin dual of G. Furthermore, let C p denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let L p (G; C p ) denote the corresponding Bochner space. If G ∋ θ ↦ A θ is the mapping belonging to L p (G; C p ), then If G is a finite group, then the previous equations comprise several generalizations of Clarkson-McCarthy inequalities obtained earlier (e.g., G = Z n or G = Z 2n ), as well as the original inequalities, for G = Z 2 . We also obtain other related inequalities.