Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm, for 2 ≤ p < ∞, and the reverse inequality holds for 0 < p ≤ 2. Moreover, we prove that if ω₀,..., ω _n−1 are the n roots of unity with ω _j = e ^2πij/n , 0 ≤ j ≤ n − 1, then for every unitarily invariant norm, for 2 ≤ p < ∞, and the reverse inequalities hold for 0 < p ≤ 2. These inequalities, which involve n-tuples of operators, lead to natural generalizations and refinements of some of the classical Clarkson inequalities in the Schatten p-norms. Extensions of these inequalities to certain convex and concave functions, including the power functions, are olso optained.